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# Failure of the curvature-dimension condition in sub-Finsler manifolds

Mattia Magnabosco111Institut für Angewandte Mathematik, Universität Bonn.
E-mail<EMAIL_ADDRESS>and Tommaso Rossi222Institut für Angewandte
Mathematik, Universität Bonn. E-mail<EMAIL_ADDRESS>
###### Abstract

The Lott–Sturm–Villani curvature-dimension condition $\mathsf{CD}(K,N)$
provides a synthetic notion for a metric measure space to have curvature
bounded from below by $K$ and dimension bounded from above by $N$. It has been
recently proved that this condition does not hold in sub-Riemannian geometry
for every choice of the parameters $K$ and $N$. In this paper, we extend this
result to the context sub-Finsler geometry, showing that the
$\mathsf{CD}(K,N)$ condition is not well-suited to characterize curvature in
this setting. Firstly, we show that this condition fails in (strict) sub-
Finsler manifolds equipped with a smooth strongly convex norm and with a
positive smooth measure. Secondly, we focus on the sub-Finsler Heisenberg
group, proving that curvature-dimension bounds can not hold also when the
reference norm is less regular, in particular when it is of class $C^{1,1}$.
The strategy for proving these results is a non-trivial adaptation of the work
of Juillet [Jui21], and it requires the introduction of new tools and ideas of
independent interest. Finally, we demonstrate the failure of the (weaker)
measure contraction property $\mathsf{MCP}(K,N)$ in the sub-Finsler Heisenberg
group, equipped with a singular strictly convex norm and with a positive
smooth measure. This result contrasts with what happens in the sub-Riemannian
Heisenberg group, which instead satisfies $\mathsf{MCP}(0,5)$.
Keywords: sub-Finsler geometry, curvature-dimension condition, Heisenberg
group.
AMS Mathematics Subject Classifications 2020: 53C23, 49J52, 58E10, 49N60.

###### Contents

  1. 1 Introduction
    1. 1.1 Curvature-dimension conditions
    2. 1.2 The curvature-dimension condition in sub-Riemannian geometry
    3. 1.3 Other curvature-dimension bounds in sub-Riemannian geometry
    4. 1.4 Sub-Finsler manifolds and Carnot groups
    5. 1.5 Main results
  2. 2 Preliminaries
    1. 2.1 The $\mathsf{CD}(K,N)$ condition
    2. 2.2 Strictly and strongly convex norms on $\mathbb{R}^{k}$
    3. 2.3 Sub-Finsler structures
  3. 3 The geometry of smooth sub-Finsler manifolds
    1. 3.1 The energy functional and the optimal control problem
    2. 3.2 Characterization of extremals and the exponential map
    3. 3.3 The end-point map
  4. 4 Failure of the $\mathsf{CD}(K,N)$ condition in smooth sub-Finsler manifolds
    1. 4.1 Construction of a geodesic without abnormal sub-segments
    2. 4.2 Regularity of the distance function
    3. 4.3 Volume contraction rate along geodesics
    4. 4.4 Proof of Theorem 1.5
  5. 5 Failure of the $\mathsf{CD}(K,N)$ condition in the sub-Finsler Heisenberg group
    1. 5.1 Convex trigonometry
    2. 5.2 Geodesics in the Heisenberg group
    3. 5.3 Failure of the $\mathsf{CD}(K,N)$ condition for $C^{1,1}$-norms
    4. 5.4 Failure of the $\mathsf{MCP}(K,N)$ condition for singular norms

## 1 Introduction

In the present paper, we address the validity of the Lott–Sturm–Villani
curvature-dimension (in short $\mathsf{CD}(K,N)$) condition in the setting of
sub-Finsler geometry. In particular, we prove that this condition can not hold
in a large class of sub-Finsler manifolds. Thus, on the one hand, this work
shows that the $\mathsf{CD}(K,N)$ condition is not well-suited to characterize
curvature in sub-Finsler geometry. On the other hand, we discuss how our
results could provide remarkable insights about the geometry of
$\mathsf{CD}(K,N)$ spaces.

### 1.1 Curvature-dimension conditions

In their groundbreaking works, Sturm [Stu06a, Stu06b] and Lott–Villani [LV09]
introduced independently a synthetic notion of curvature-dimension bounds for
non-smooth spaces, using Optimal Transport. Their theory stems from the
crucial observation that, in the Riemannian setting, having a uniform lower
bound on the Ricci curvature and an upper bound on the dimension, can be
equivalently characterized in terms of a convexity property of suitable
entropy functionals in the Wasserstein space. In particular, it was already
observed in [vRS05] that the Ricci bound $\mathrm{Ric}\geq K\cdot g$ holds if
and only if the Boltzmann–Shannon entropy functional is $K$-convex in the
Wasserstein space. More generally, let $(M,g)$ be a complete Riemannian
manifold, equipped with a measure of the form $\mathfrak{m}=e^{-V}{\rm
vol}_{g}$, where ${\rm vol}_{g}$ is the Riemannian volume and $V\in C^{2}(M)$.
Given $K\in\mathbb{R}$ and $N\in(n,+\infty]$, Sturm [Stu06b] proved that the
(generalized) Ricci lower bound

$\mathrm{Ric}_{N,V}:=\mathrm{Ric}+\nabla^{2}V-\frac{\nabla V\otimes\nabla
V}{N-n}\geq K\cdot g,$ (1)

holds if and only if a $(K,N)$-convexity inequality holds for Rényi entropy
functionals, defined with respect to the reference measure $\mathfrak{m}$.
While (1) involves a differential object, the Ricci tensor, entropy convexity
can be formulated relying solely upon a reference distance and a reference
measure, without the need of the underlying smooth structure of the Riemannian
manifold. Therefore, it can be introduced in the non-smooth setting of metric
measure spaces and taken as definition of curvature-dimension bound. This
condition is called $\mathsf{CD}(K,N)$ and represents a synthetic lower bound
on the (Ricci) curvature by $K\in\mathbb{R}$ and a synthetic upper bound on
the dimension by $N\in(1,\infty]$, see Definition 2.2. In this sense,
according to the discussion above, the $\mathsf{CD}(K,N)$ condition is
coherent with the Riemannian setting. Moreover, it was proved by Ohta [Oht09]
that the relation between curvature and $\mathsf{CD}(K,N)$ condition holds
also in the context of Finsler manifolds.

Remarkably, $\mathsf{CD}(K,N)$ spaces (i.e. spaces satisfying the
$\mathsf{CD}(K,N)$ condition) enjoy several geometric properties which hold in
the smooth setting. Some of them are expected (and in a way necessary) for a
reasonable curvature-dimension bound, such as the scaling [Stu06b],
tensorization [BS10] and globalization [CM21] properties or the monotonicity
with respect to the parameters [Stu06b], i.e.

$\mathsf{CD}(K^{\prime},N^{\prime})\implies\mathsf{CD}(K,N)\qquad\text{if
}K^{\prime}\geq K\text{ and }N^{\prime}\leq N.$

Others are completely non-trivial and highlight some notable geometric
features. Among them, we mention the Bonnet–Myers diameter bound and the
Bishop–Gromov inequality, that provides an estimate on the volume growth of
concentric balls. Particularly interesting in the context of this work is the
Brunn–Minkowski inequality $\mathsf{BM}(K,N)$, which, given two sets $A$ and
$B$ in the reference metric measure space
$(\mathsf{X},\mathsf{d},\mathfrak{m})$, provides a lower estimate on the
measure of the set of $t$-midpoints

$M_{t}(A,B)=\left\\{x\in\mathsf{X}\,:\,\mathsf{d}(a,x)=t\mathsf{d}(a,b),\,\mathsf{d}(x,b)=(1-t)\mathsf{d}(a,b)\,\text{
for some }a\in A,b\in B\right\\},$ (2)

in terms of $\mathfrak{m}(A)$ and $\mathfrak{m}(B)$, for every $t\in[0,1]$,
cf. (12). The notable feature of the $\mathsf{BM}(K,N)$ inequality is that its
formulation does not invoke optimal transport, or Wasserstein interpolation,
and because of that, it is easier to handle than the $\mathsf{CD}(K,N)$
condition. Nonetheless, it contains a strong information about the curvature
of the underlying space, to the extent that it is equivalent to the
$\mathsf{CD}(K,N)$ condition in the Riemannian setting, cf. [MPR22a]. In
particular, in the proof of Theorem 1.5 and Theorem 1.7, we show the failure
of the $\mathsf{CD}(K,N)$ condition by contradicting the Brunn–Minkowski
inequality $\mathsf{BM}(K,N)$.

Finally, another fundamental property of the $\mathsf{CD}(K,N)$ condition is
its stability with respect to the (pointed) measured Gromov–Hausdorff
convergence [Stu06b, LV09, GMS13]. This notion of convergence for metric
measure spaces essentially combines the Hausdorff convergence for the metric
side and the weak convergence for the reference measures. As in a metric
measure space, the tangent spaces at a point are identified with a measured
Gromov–Hausdorff limit procedure of suitably rescalings of the original space,
the stability of the curvature-dimension condition implies that the metric
measure tangents of a $\mathsf{CD}(K,N)$ space is a $\mathsf{CD}(0,N)$ space.

In the setting of metric measure spaces, it is possible to define other
curvature-dimension bounds, such as the so-called measure contraction property
(in short $\mathsf{MCP}(K,N)$), introduced by Ohta in [Oht07]. In broad terms,
the $\mathsf{MCP}(K,N)$ condition can be interpreted as the Brunn–Minkowski
inequality where one of the two sets degenerates to a point. In particular, it
is implied by (and strictly weaker than) the $\mathsf{BM}(K,N)$ inequality,
and therefore it is also a consequence of the $\mathsf{CD}(K,N)$ condition.

### 1.2 The curvature-dimension condition in sub-Riemannian geometry

While in the Riemannian setting the $\mathsf{CD}(K,N)$ condition is equivalent
to having bounded geometry, a similar result does not hold for sub-Riemannian
manifolds. Sub-Riemannian geometry is a generalization of Riemannian geometry
where, given a smooth manifold $M$, we define a smoothly varying scalar
product only on a subset of _horizontal_ directions $\mathcal{D}_{p}\subset
T_{p}M$ (called distribution) at each point $p\in M$. Under the so-called
Hörmander condition, $M$ is horizontally-path connected, and the usual length-
minimization procedure yields a well-defined distance $\mathsf{d}_{SR}$. In
particular, differently from what happens in Riemannian geometry, the rank of
the distribution $r(p):=\dim\mathcal{D}_{p}$ may be strictly less than the
dimension of the manifold and may vary with the point. This may influence the
behavior of geodesics, emphasizing singularities of the distance
$\mathsf{d}_{SR}$. For this reason, we can not expect the $\mathsf{CD}(K,N)$
condition to hold for _truly_ sub-Riemannian manifolds. This statement is
confirmed by a series of papers, most notably [Jui21, MR23, RS23], that
contributed to the proof of the following result (see also [AS20] for the
analogous result in Carnot groups).

######  Theorem 1.1.

Let $M$ be a complete truly sub-Riemannian manifold, equipped with a positive
smooth measure $\mathfrak{m}$. Then, the metric measure space
$(M,\mathsf{d}_{SR},\mathfrak{m})$ does not satisfy the $\mathsf{CD}(K,N)$
condition, for any $K\in\mathbb{R}$ and $N\in(1,\infty)$.

In [Jui21], Juillet proved Theorem 1.1 for sub-Riemannian manifolds where the
rank of the distribution $r(p)$ is strictly smaller than the topological
dimension $n:=\dim M$, for every $p\in M$. His strategy relies on the
construction of two Borel subsets for which the Brunn–Minkowski inequality
$\mathsf{BM}(K,N)$ does not hold. Namely, for all $R,\varepsilon>0$, one can
find $A,B\subset M$ such that $\mathrm{diam}(A\cup B)<R$,
$\mathfrak{m}(A)\approx\mathfrak{m}(B)$, and such that there exists
$t\in(0,1)$ for which

$\mathfrak{m}(M_{t}(A,B))\leq\frac{1}{2^{\mathcal{N}-n}}\,\mathfrak{m}(B)(1+\varepsilon),$
(3)

where $\mathcal{N}$ is the so-called _geodesic dimension_ of $M$, see [ABR18,
Def. 5.47] for a precise definition. The sets $A$ and $B$ are metric balls of
small radius, centered at the endpoints of a short segment of an _ample
geodesic_ , see [ABR18] for details. The inequality (3) allows to contradict
the Brunn–Minkowski inequality $\mathsf{BM}(K,N)$ if and only if the geodesic
dimension $\mathcal{N}$ is strictly greater than $n$, which is the case if
$r(p)<n$, for every $p\in M$.

While Julliet’s result is quite general, it does not include almost-Riemannian
geometry. Roughly speaking, an almost-Riemannian manifold is a sub-Riemannian
manifold where the rank of the distribution coincides with the dimension of
$M$, at almost every point. In [MR23], we addressed this issue, proposing a
new strategy for proving Theorem 1.1 in this setting. Our idea is to exploit
the following one-dimensional characterization of the $\mathsf{CD}(K,N)$
condition:

$\mathsf{CD}(K,N)\quad\Rightarrow\quad\mathsf{CD}^{1}(K,N),$ (4)

proved by Cavalletti and Mondino in [CM17], and contradict the
$\mathsf{CD}^{1}(K,N)$ condition. On a metric measure space
$(\mathsf{X},\mathsf{d},\mathfrak{m})$, given a $1$-Lipschitz function
$u\in\mathsf{Lip}(\mathsf{X})$, it is possible to partition $\mathsf{X}$ in
one-dimensional transport rays, associated with $u$, and disintegrate the
measure $\mathfrak{m}$ accordingly. Then, the $\mathsf{CD}^{1}(K,N)$ condition
asks for the validity of the $\mathsf{CD}(K,N)$ condition along the transport
rays of the disintegration associated with $u$, for any choice of
$u\in\mathsf{Lip}(\mathsf{X})$. In [MR23, Thm. 1.2], when $M$ is either
strongly regular or $\dim M=2$, we are able to explicitly build a
$1$-Lipschitz function, and compute the associated disintegration, showing
that the $\mathsf{CD}(K,N)$ condition along the rays does not hold for any
$K\in\mathbb{R}$ and $N\in(1,\infty)$.

Most recently, Rizzi and Stefani [RS23] proposed yet another strategy to prove
Theorem 1.1. Differently from the strategies presented above, they pursue the
“Eulerian” approach to curvature-dimension bounds, based on a suitable Gamma
calculus, see [BGL14] for details. This approach can be adopted for metric
measure spaces that satisfy the _infinitesimal Hibertian_ condition (cf.
[AGS14, Gig15]) which forces the space to be _Riemannian-like_ and ensures the
linearity of the heat flow. According to [AGS15], an infinitesimally
Hilbertian $\mathsf{CD}(K,N)$ space supports the so-called Bakry–Émery
inequality $\mathsf{BE}(K,\infty)$, which, in the sub-Riemannian setting reads
as

$\frac{1}{2}\Delta\left(\|\nabla f\|^{2}\right)\geq g(\nabla f,\nabla\Delta
f)+K\|\nabla f\|^{2},\qquad\forall\,f\in C^{\infty}_{c}(M),$ (5)

where $\nabla$ is the horizontal gradient and $\Delta$ is the sub-Laplacian.
In [RS23], the authors show that (5) implies the existence of enough
isometries on the metric tangent to force it to be Euclidean at _each point_ ,
proving Theorem 1.1 (including also the case $N=\infty$).

### 1.3 Other curvature-dimension bounds in sub-Riemannian geometry

Given that the $\mathsf{CD}(K,N)$ condition does not hold in sub-Riemannian
geometry, considerable efforts have been undertaken to explore potential
curvature-dimension bounds that may hold in this class. A first observation in
this direction is that the weaker $\mathsf{MCP}(K,N)$ condition does hold in
many examples of sub-Riemannian manifolds. In particular, it was proved by
Juillet [Jui09] that the sub-Riemannian Heisenberg group satisfies the
$\mathsf{MCP}(0,5)$ condition, where the curvature-dimension parameters can
not be improved. Moreover, in [ABR18] it was observed that the optimal
dimensional parameter for the measure contraction property coincides with the
geodesic dimension of the sub-Riemannian Heisenberg group (i.e.
$\mathcal{N}=5$). This result has been subsequently extended to a large class
of sub-Riemannian manifolds, including ideal Carnot groups [Rif13], corank-$1$
Carnot groups [Riz16], generalised H-type Carnot groups [BR18] and two-step
analytic sub-Riemannian structures [BR20]. In all these cases, the
$\mathsf{MCP}(0,N)$ condition holds with the dimensional parameter $N$ greater
than or equal to the geodesic dimension $\mathcal{N}$.

Another attempt is due to Milman [Mil21], who introduced the quasi curvature-
dimension condition, inspired by the interpolation inequalities along
Wasserstein geodesics in ideal sub-Riemannian manifolds, proved by Barilari
and Rizzi [BR19]. Finally, these efforts culminated in the recent work by
Barilari, Mondino and Rizzi [BMR22], where the authors propose a unification
of Riemannian and sub-Riemannian geometries in a comprehensive theory of
synthetic Ricci curvature lower bounds. In the setting of gauge metric measure
spaces, they introduce the $\mathsf{CD}(\beta,n)$ condition, encoding in the
distortion coefficient $\beta$ finer geometrical information of the underlying
structure. Moreover they prove that the $\mathsf{CD}(\beta,n)$ condition holds
for compact fat sub-Riemannian manifolds, thus substantiating the definition.

### 1.4 Sub-Finsler manifolds and Carnot groups

In the present paper, we focus on sub-Finsler manifolds, which widely
generalize both sub-Riemannian and Finsler geometry. Indeed, in this setting,
given a smooth manifold $M$, we prescribe a smoothly varying _norm_ (which
needs not be induced by a scalar product) on the distribution
$\mathcal{D}_{p}\subset T_{p}M$, at each point $p\in M$. As in the sub-
Riemannian setting, $\mathcal{D}$ must satisfy the Hörmander condition, and
consequently the length-minimization procedure among admissible curves gives a
well-defined distance $\mathsf{d}_{SF}$. Note that, on the one hand, if the
$\mathcal{D}_{p}=T_{p}M$ for every $p\in M$, we recover the classical Finsler
geometry. On the other hand, if the norm on $\mathcal{D}_{p}$ is induced by a
scalar product for every $p\in M$, we fall back into sub-Riemannian geometry.

Replacing the scalar product with a (possibly singular) norm is not merely a
technical choice, as the metric structure of a sub-Finsler manifold reflects
the singularities of the reference norm. Indeed, even though sub-Finsler
manifolds can still be investigated by means of classical control theory
[AS04], deducing finer geometrical properties is more delicate compared to
what happens in the sub-Riemannian setting, as the Hamiltonian function has a
low regularity, cf. Section 3. In this regard, sub-Finsler manifolds provide
an interesting example of smooth structures which present both the typical
sub-Riemannian and Finsler singular behavior. A particularly relevant class of
sub-Finsler manifolds is the one of _sub-Finsler Carnot groups_.

######  Definition 1.2 (Carnot group).

A Carnot group is a connected, simply connected Lie group $G$ with nilpotent
Lie algebra $\mathfrak{g}$, admitting a stratification

$\mathfrak{g}=\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{k},$

where $\mathfrak{g}_{i+1}=[\mathfrak{g}_{1},\mathfrak{g}_{i}]$, for every
$i=1,\dots,k-1$, and $[\mathfrak{g}_{1},\mathfrak{g}_{k}]=\\{0\\}$.

Given a Carnot group $G$, if we equip the first layer $\mathfrak{g}_{1}$ of
its Lie algebra with a norm, we naturally obtain a left-invariant sub-Finsler
structure on $G$. We refer to the resulting manifold as a sub-Finsler Carnot
group.

Motivated from the results presented in the previous section, cf. Theorem 1.1,
and especially from the ones obtained in the present work (see Section 1.5),
we formulate the following conjecture.

######  Conjecture 1.3.

Let $G$ be a sub-Finsler Carnot group, endowed with a positive smooth measure
$\mathfrak{m}$. Then, the metric measure space
$(G,\mathsf{d}_{SF},\mathfrak{m})$ does not satisfy the $\mathsf{CD}(K,N)$
condition for any $K\in\mathbb{R}$ and $N\in(1,\infty)$.

Our interest in Carnot groups stems from the fact that they are the only
metric spaces that are locally compact, geodesic, isometrically homogeneous
and self-similar (i.e. admitting a dilation) [LD15]. According to this
property, sub-Finsler Carnot groups naturally arise as metric tangents of
metric measure spaces.

######  Theorem 1.4 (Le Donne [LD11]).

Let $(\mathsf{X},\mathsf{d},\mathfrak{m})$ be a geodesic metric measure space,
equipped with a doubling measure $\mathfrak{m}$. Assume that, for
$\mathfrak{m}$-almost every $x\in\mathsf{X}$, the set ${\rm
Tan}(\mathsf{X},x)$ of all metric tangent spaces at $x$ contains only one
element. Then, for $\mathfrak{m}$-almost every $x\in\mathsf{X}$, the element
in ${\rm Tan}(\mathsf{X},x)$ is a sub-Finsler Carnot group $G$.

In particular, this result applies to $\mathsf{CD}(K,N)$ spaces, where the
validity of the doubling property is guaranteed by the Bishop–Gromov
inequality. Moreover, as already mentioned, the metric measure tangents of a
$\mathsf{CD}(K,N)$ space are $\mathsf{CD}(0,N)$. Therefore, the study of the
$\mathsf{CD}(K,N)$ condition in sub-Finsler Carnot groups, and especially the
validity of Conjecture 1.3, has the potential to provide deep insights on the
structure of tangents of $\mathsf{CD}(K,N)$ spaces. This could be of
significant interest, particularly in connection with Bate’s recent work
[Bat22], which establishes a criterion for rectifiability in metric measure
spaces, based on the structure of metric tangents.

### 1.5 Main results

The aim of this paper is to show the failure of the $\mathsf{CD}(K,N)$
condition in the sub-Finsler setting, with a particular attention to
Conjecture 1.3. Our results offer an advance into two different directions: on
the one hand we deal with general sub-Finsler structures, where the norm is
smooth, cf. Theorem 1.5 and Theorem 1.6, and, on the other hand, we deal with
the sub-Finsler Heisenberg group, equipped with more general norms, cf.
Theorem 1.8 and Theorem 1.7.

In order to extend the sub-Riemannian result of Theorem 1.1 to the sub-Finsler
setting, one can attempt to adapt the strategies discussed in Section 1.2,
however this can present major difficulties. Specifically, the argument
developed in [RS23] has little hope to be generalized, because the
infinitesimal Hilbertianity assumption does not hold in Finsler-like spaces,
see [OS12]. It is important to note that this is not solely a “regularity”
issue, in the sense that it also occurs when the norm generating the sub-
Finsler structure is smooth, but not induced by a scalar product. Instead, the
approach proposed in [MR23] could potentially be applied to sub-Finsler
manifolds as it relies on tools developed in the non-smooth setting, see (4).
However, adapting the sub-Riemannian computations that led to a contradiction
of the $\mathsf{CD}^{1}(K,N)$ condition seems non-trivial already when the
reference norm is smooth. Finally, the strategy illustrated in [Jui21] hinges
upon geometrical constructions and seems to be well-suited to generalizations
to the sub-Finsler setting. In this paper, we build upon this observation and
adapt the latter strategy to prove our main theorems.

Our first result is about the failure of the $\mathsf{CD}(K,N)$ condition in
_smooth_ sub-Finsler manifolds, cf. Theorem 4.26.

######  Theorem 1.5.

Let $M$ be a complete sub-Finsler manifold with $r(p)<n:=\dim M$ for every
$p\in M$, equipped with a smooth, strongly convex norm ${\|\\!\cdot\\!\|}$ and
with a positive smooth measure $\mathfrak{m}$. Then, the metric measure space
$(M,\mathsf{d}_{SF},\mathfrak{m})$ does not satisfy the $\mathsf{CD}(K,N)$
condition, for any $K\in\mathbb{R}$ and $N\in(1,\infty)$.

This result is the sub-Finsler analogue of [Jui21, Cor. 1.2]. Although the
strategy of its proof follows the blueprint of [Jui21], the adaptation to our
setting is non-trivial and requires many intermediate results of independent
interest. First of all, we establish the existence of geodesics _without
abnormal sub-segments_ , cf. Theorem 4.11, proposing a construction that is
new even in the sub-Riemannian framework and relies on the regularity
properties of the distance function from the boundary of an open set. Note
that, while these properties are well-known in the sub-Riemannian context (cf.
[FPR20, Prop. 3.1]), inferring them in the sub-Finsler setting becomes more
challenging due to the low regularity of the Hamiltonian, which affects the
regularity of the normal exponential map. Nonetheless, we settle a weaker
regularity result that is enough for our purposes, cf. Theorem 4.8. Second of
all, we prove an analogue of the sub-Riemannian theorem, indicating that the
volume contraction along ample geodesics is governed by the geodesic
dimension, see [ABR18, Thm. D]. Indeed, in a smooth sub-Finsler manifold, we
establish that the volume contraction rate along geodesics without abnormal
sub-segments is bigger than $\dim M+1$, cf. Theorem 4.22. Finally, we mention
that these technical challenges lead us to a simplification of Juillet’s
argument (cf. Theorem 4.26), which revealed itself to be useful also in the
proof of Theorem 1.7.

Observe that, since sub-Finsler Carnot groups are equiregular (and thus
$r(p)<n$, for every $p\in G$) and complete, we immediately obtain the
following consequence of Theorem 1.5, which constitutes a significant step
forward towards the proof of Conjecture 1.3.

######  Theorem 1.6.

Let $G$ be a sub-Finsler Carnot group, equipped with a smooth, strongly convex
norm ${\|\\!\cdot\\!\|}$ and with a positive smooth measure $\mathfrak{m}$.
Then, the metric measure space $(G,\mathsf{d}_{SF},\mathfrak{m})$ does not
satisfy the $\mathsf{CD}(K,N)$ condition, for any $K\in\mathbb{R}$ and
$N\in(1,\infty)$.

In the proof of Theorem 1.5, the smoothness of the norm plays a pivotal role
in establishing the correct volume contraction rate along geodesics. When the
norm is less regular, it is not clear how to achieve an analogue behavior in
full generality. Nonetheless, we are able to recover such a result in the
context of the _sub-Finsler Heisenberg group_ $\mathbb{H}$, equipped with a
possibly singular norm (see Section 5). Working in this setting is
advantageous since, assuming strict convexity of the norm, the geodesics and
the cut locus are completely described [Ber94] and there exists an explicit
expression for them in terms of convex trigonometric functions [Lok21] (see
also [BBLDS17] for an example of the non-strictly convex case).

For the sub-Finsler Heisenberg group, we prove two different results, with the
first addressing the case of $C^{1,1}$ and strictly convex reference norms and
thus substantially relaxing the assumptions of Theorem 1.5, cf. Theorem 5.23.

######  Theorem 1.7.

Let $\mathbb{H}$ be the sub-Finsler Heisenberg group, equipped with a strictly
convex and $C^{1,1}$ norm and with a positive smooth measure $\mathfrak{m}$.
Then, the metric measure space $(\mathbb{H},\mathsf{d}_{SF},\mathfrak{m})$
does not satisfy the $\mathsf{CD}(K,N)$, for any $K\in\mathbb{R}$ and
$N\in(1,\infty)$.

The proof of this statement follows the same lines of [Jui21, Cor. 1.2].
However, the low regularity of the norm, and thus of geodesics, prevent us to
exploit the same differential tools developed for Theorem 1.5. Nonetheless,
using the explicit expression of geodesics and of the exponential map, we can
still recover an analogue result. In particular, guided by the intuition that
a contraction rate along geodesics, similar to the one appearing in the smooth
case, should still hold, we thoroughly study the Jacobian determinant of the
exponential map. Building upon a fine analysis of convex trigonometric
functions, cf. Section 5.1 and Proposition 5.12 where the regularity of the
norm plays a pivotal role, we obtain an estimate on the contraction rate of
the Jacobian determinant of the exponential map. However, this behavior is
determined only for a large (in a measure-theoretic sense) set of covectors in
the cotangent space. This poses additional challenges that we are able to
overcome with a delicate density-type argument, together with an extensive use
of the left-translations of the group, cf. Theorem 5.23 and also Remark 5.24.
Remarkably, for every $C^{1,1}$ reference norm, we obtain the exact same
contraction rate, equal to the geodesic dimension $\mathcal{N}=5$, that
characterizes the sub-Riemannian Heisenberg group. Finally, let us mention
that, recently, Borza and Tashiro [BT23] showed that the Heisenberg group
equipped with the $l^{p}$-norm can not satisfy the $\mathsf{MCP}(K,N)$
condition if $p>2$. In particular, $l^{p}$-norms with $p>2$ are strictly
convex $C^{1,1}$ norms, hence their result implies Theorem 1.7 for those
specific cases. However, we stress that their technique is based on explicit
computations that can not be carried out in our general setting.

Our second result in the sub-Finsler Heisenberg group deals with the case of
singular (i.e. non-$C^{1}$) reference norms, cf. Theorem 5.25.

######  Theorem 1.8.

Let $\mathbb{H}$ be the sub-Finsler Heisenberg group, equipped with a strictly
convex norm ${\|\\!\cdot\\!\|}$ which is not $C^{1}$, and let $\mathfrak{m}$
be a positive smooth measure on $\mathbb{H}$. Then, the metric measure space
$(\mathbb{H},\mathsf{d}_{SF},\mathfrak{m})$ does not satisfy the measure
contraction property $\mathsf{MCP}(K,N)$ for any $K\in\mathbb{R}$ and
$N\in(1,\infty)$.

Observe that this theorem also shows the failure of the $\mathsf{CD}(K,N)$
condition, which is stronger than the measure contraction property
$\mathsf{MCP}(K,N)$. However, Theorem 1.8 has an interest that goes beyond
this consequence, as it reveals a phenomenon that stands in contrast to what
typically happens in the sub-Riemannian setting. In fact, as already mentioned
in section 1.3, the $\mathsf{MCP}(K,N)$ condition holds in many sub-Riemannian
manifolds, and, in the particular case of the sub-Riemannian Heisenberg group,
holds with (sharp) parameters $K=0$ and $N=5$. Therefore, Theorem 1.8 shows
that a singularity of the reference norm can cause the failure of the measure
contraction property $\mathsf{MCP}(K,N)$.

Our strategy to show Theorem 1.8 consists in finding a set
$A\subset\mathbb{H}$, having positive $\mathfrak{m}$-measure, such that the
set of $t$-midpoints $M_{t}(\\{{\rm e}\\},A)$ (where ${\rm e}$ denotes the
identity in $\mathbb{H}$) is $\mathfrak{m}$-null for every $t$ sufficiently
small. This construction is based on a remarkable geometric property of the
space $(\mathbb{H},\mathsf{d}_{SF},\mathfrak{m})$, where geodesics can branch,
even though they are unique. This has independent interest, as examples of
branching spaces usually occur when geodesics are not unique.

We conclude this section highlighting that the combination of Theorem 1.7 and
Theorem 1.8 proves Conjecture 1.3 for a large class of sub-Finsler Heisenberg
groups. This is particularly interesting as the sub-Finsler Heisenberg groups
are the unique sub-Finsler Carnot groups with Hausdorff dimension less than
$5$ (or with topological dimension less than or equal to $3$), up to
isometries.

### Structure of the paper

In Section 2 we introduce all the necessary preliminaries. In particular, we
present the precise definition of the $\mathsf{CD}(K,N)$ condition with some
of its consequences, and we introduce the notion of sub-Finsler structure on a
manifold. Section 3 is devoted to the study of the geometry of sub-Finsler
manifolds. For the sake of completeness, we include generalizations of various
sub-Riemannian results, especially regarding the characterizations of normal
and abnormal extremals and the exponential map. In Section 4, we present the
proof of Theorem 1.5. We start by developing the building blocks for it,
namely the existence of a geodesic without abnormal sub-segments and the
regularity of the distance function. Then, we estimate the volume contraction
rate, along the previously selected geodesic. Finally, in Section 4.4, we
adapt Juillet’s strategy to obtain our first main theorem. Section 5 collects
our results about the failure of the $\mathsf{CD}(K,N)$ condition in the sub-
Finsler Heisenberg group. After having introduced the convex trigonometric
functions in Section 5.1, we use them to provide the explicit explicit
expression of geodesics, cf. Section 5.2. We conclude by proving Theorem 1.7
in Section 5.3 and Theorem 1.8 in Section 5.4.

### Acknowledgments

T.R. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through the collaborative research centre “The
mathematics of emerging effects” (CRC 1060, Project-ID 211504053). The autors
wish to thank Luca Rizzi for stimulating discussions regarding the regularity
of the distance function in sub-Finsler manifolds and Lorenzo Portinale for
his careful reading of the introduction.

## 2 Preliminaries

### 2.1 The $\mathsf{CD}(K,N)$ condition

A metric measure space is a triple $(\mathsf{X},\mathsf{d},\mathfrak{m})$
where $(\mathsf{X},\mathsf{d})$ is a complete and separable metric space and
$\mathfrak{m}$ is a locally finite Borel measure on it. In the following
$C([0,1],\mathsf{X})$ will stand for the space of continuous curves from
$[0,1]$ to $\mathsf{X}$. A curve $\gamma\in C([0,1],\mathsf{X})$ is called
geodesic if

$\mathsf{d}(\gamma(s),\gamma(t))=|t-s|\cdot\mathsf{d}(\gamma(0),\gamma(1))\quad\text{for
every }s,t\in[0,1],$ (6)

and we denote by $\operatorname{Geo}(\mathsf{X})$ the space of geodesics on
$\mathsf{X}$. The metric space $(\mathsf{X},\mathsf{d})$ is said to be
geodesic if every pair of points $x,y\in\mathsf{X}$ can be connected with a
curve $\gamma\in\operatorname{Geo}(\mathsf{X})$. For any $t\in[0,1]$ we define
the evaluation map $e_{t}\colon C([0,1],\mathsf{X})\to\mathsf{X}$ by setting
$e_{t}(\gamma):=\gamma(t)$. We denote by $\mathscr{P}(\mathsf{X})$ the set of
Borel probability measures on $\mathsf{X}$ and by
$\mathscr{P}_{2}(\mathsf{X})\subset\mathscr{P}(\mathsf{X})$ the set of those
having finite second moment. We endow the space $\mathscr{P}_{2}(\mathsf{X})$
with the Wasserstein distance $W_{2}$, defined by

$W_{2}^{2}(\mu_{0},\mu_{1}):=\inf_{\pi\in\mathsf{Adm}(\mu_{0},\mu_{1})}\int\mathsf{d}^{2}(x,y)\,\,\mathrm{d}\pi(x,y),$
(7)

where $\mathsf{Adm}(\mu_{0},\mu_{1})$ is the set of all the admissible
transport plans between $\mu_{0}$ and $\mu_{1}$, namely all the measures in
$\mathscr{P}(\mathsf{X}\times\mathsf{X})$ such that
$(\mathtt{p}_{1})_{\sharp}\pi=\mu_{0}$ and
$(\mathtt{p}_{2})_{\sharp}\pi=\mu_{1}$. The metric space
$(\mathscr{P}_{2}(\mathsf{X}),W_{2})$ is itself complete and separable,
moreover, if $(\mathsf{X},\mathsf{d})$ is geodesic, then
$(\mathscr{P}_{2}(\mathsf{X}),W_{2})$ is geodesic as well. In particular,
every geodesic $(\mu_{t})_{t\in[0,1]}$ in
$(\mathscr{P}_{2}(\mathsf{X}),W_{2})$ can be represented with a measure
$\eta\in\mathscr{P}(\operatorname{Geo}(\mathsf{X}))$, meaning that
$\mu_{t}=(e_{t})_{\\#}\eta$.

We are now ready to introduce the $\mathsf{CD}(K,N)$ condition, pioneered by
Sturm and Lott–Villani [Stu06a, Stu06b, LV09]. As already mentioned, this
condition aims to generalize, to the context metric measure spaces, the notion
of having Ricci curvature bounded from below by $K\in\mathbb{R}$ and dimension
bounded above by $N>1$. In order to define the $\mathsf{CD}(K,N)$ condition,
let us introduce the following distortion coefficients: for every
$K\in\mathbb{R}$ and $N\in(1,\infty)$,

$\tau_{K,N}^{(t)}(\theta):=t^{\frac{1}{N}}\left[\sigma_{K,N-1}^{(t)}(\theta)\right]^{1-\frac{1}{N}},$
(8)

where

$\sigma_{K,N}^{(t)}(\theta):=\begin{cases}\displaystyle\frac{\sin(t\theta\sqrt{K/N})}{\sin(\theta\sqrt{K/N})}&\textrm{if}\
N\pi^{2}>K\theta^{2}>0,\crcr t&\textrm{if}\
K=0,\crcr\displaystyle\frac{\sinh(t\theta\sqrt{-K/N})}{\sinh(\theta\sqrt{-K/N})}&\textrm{if}\
K<0.\end{cases}$ (9)

######  Remark 2.1.

Observe that for every $K\in\mathbb{R}$, $N\in(1,\infty)$ and $t\in[0,1]$ we
have

$\lim_{\theta\to 0}\sigma_{K,N}^{(t)}(\theta)=t\qquad\text{ and
}\qquad\lim_{\theta\to 0}\tau_{K,N}^{(t)}(\theta)=t.$

######  Definition 2.2.

A metric measure space $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is said to be a
$\mathsf{CD}(K,N)$ space (or to satisfy the $\mathsf{CD}(K,N)$ condition) if
for every pair of measures
$\mu_{0}=\rho_{0}\mathfrak{m},\mu_{1}=\rho_{1}\mathfrak{m}\in\mathscr{P}_{2}(\mathsf{X})$,
absolutely continuous with respect to $\mathfrak{m}$, there exists a
$W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ connecting them and induced by
$\eta\in\mathscr{P}(\operatorname{Geo}(\mathsf{X}))$, such that for every
$t\in[0,1]$, $\mu_{t}=\rho_{t}\mathfrak{m}\ll\mathfrak{m}$ and the following
inequality holds for every $N^{\prime}\geq N$ and every $t\in[0,1]$

$\int_{\mathsf{X}}\rho_{t}^{1-\frac{1}{N^{\prime}}}\,\mathrm{d}\mathfrak{m}\geq\int_{\mathsf{X}\times\mathsf{X}}\Big{[}\tau^{(1-t)}_{K,N^{\prime}}\big{(}\mathsf{d}(x,y)\big{)}\rho_{0}(x)^{-\frac{1}{N^{\prime}}}+\tau^{(t)}_{K,N^{\prime}}\big{(}\mathsf{d}(x,y)\big{)}\rho_{1}(y)^{-\frac{1}{N^{\prime}}}\Big{]}\,\mathrm{d}\pi(x,y),$
(10)

where $\pi=(e_{0},e_{1})_{\\#}\eta$.

One of the most important merits of the $\mathsf{CD}(K,N)$ condition is that
it is sufficient to deduce geometric and functional inequalities that hold in
the smooth setting. An example which is particularly relevant for this work is
the so-called Brunn–Minkowski inequality, whose definition in the metric
measure setting requires the following notion.

######  Definition 2.3.

Let $(\mathsf{X},\mathsf{d})$ be a metric space and let $A,B\subset\mathsf{X}$
be two Borel subsets. Then for $t\in(0,1)$, we defined the set of
$t$-_midpoints_ between $A$ and $B$ as

$\displaystyle
M_{t}(A,B):=\\{x\in\mathsf{X}\,:\,x=\gamma(t)\,,\,\gamma\in\operatorname{Geo}(\mathsf{X})\,,\,\gamma(0)\in
A\,,\ \text{and}\ \gamma(1)\in B\\}\,.$ (11)

We can now introduce the metric measure version of the Brunn–Minkowski
inequality, whose formulation is stated in terms of the distortion
coefficients (8).

######  Definition 2.4.

Given $K\in\mathbb{R}$ and $N\in(1,\infty)$, we say that a metric measure
space $(\mathsf{X},\mathsf{d},\mathfrak{m})$ satisfies the _Brunn–Minkowski
inequality_ $\mathsf{BM}(K,N)$ if, for every nonempty
$A,B\subset\text{spt}(\mathfrak{m})$ Borel subsets, $t\in(0,1)$, we have

$\displaystyle\mathfrak{m}\big{(}M_{t}(A,B)\big{)}\big{)}^{\frac{1}{N}}\geq\tau_{K,N}^{(1-t)}(\Theta(A,B))\cdot\mathfrak{m}(A)^{\frac{1}{N}}+\tau_{K,N}^{(t)}(\Theta(A,B))\cdot\mathfrak{m}(B)^{\frac{1}{N}}\,,$
(12)

where

$\displaystyle\Theta(A,B):=\left\\{\begin{array}[]{ll}\displaystyle\inf_{x\in
A,\,y\in B}\mathsf{d}(x,y)&\text{ if }K\geq 0\,,\\\ \displaystyle\sup_{x\in
A,\,y\in B}\mathsf{d}(x,y)&\text{ if }K<0\,.\end{array}\right.$ (15)

As already mentioned, the Brunn–Minkowski inequality is a consequence of the
$\mathsf{CD}(K,N)$ condition, in particular we have that

$\mathsf{CD}(K,N)\implies\mathsf{BM}(K,N),$

for every $K\in\mathbb{R}$ and every $N\in(1,\infty)$. In Sections 4 and 5, we
are going to disprove the $\mathsf{CD}(K,N)$ condition for every choice of the
parameters $K\in\mathbb{R}$ and $N\in(1,\infty)$, by contradicting the
Brunn–Minkowski inequality $\mathsf{BM}(K,N)$. A priori, this is a stronger
result than the ones stated in Theorem 1.5 and Theorem 1.7, since the
Brunn–Minkowski inequality is (in principle) weaker than the
$\mathsf{CD}(K,N)$ condition. However, recent developments (cf. [MPR22a,
MPR22b]) suggest that the Brunn–Minkowski $\mathsf{BM}(K,N)$ could be
equivalent to the $\mathsf{CD}(K,N)$ condition in a wide class of metric
measure spaces.

Another curvature-dimension bound, which can be defined for metric measure
spaces, is the so-called measure contraction property (in short
$\mathsf{MCP}(K,N)$), that was introduced by Ohta in [Oht07]. The idea behind
it is basically to require the $\mathsf{CD}(K,N)$ condition to hold when the
first marginal degenerates to $\delta_{x}$, a delta-measure at
$x\in\text{spt}(\mathfrak{m})$, and the second marginal is
$\frac{\mathfrak{m}|_{A}}{\mathfrak{m}(A)}$, for some Borel set
$A\subset\mathsf{X}$ with $0<\mathfrak{m}(A)<\infty$.

######  Definition 2.5 ($\mathsf{MCP}(K,N)$ condition).

Given $K\in\mathbb{R}$ and $N\in(1,\infty)$, a metric measure space
$(\mathsf{X},\mathsf{d},\mathfrak{m})$ is said to satisfy the _measure
contraction property_ $\mathsf{MCP}(K,N)$ if for every
$x\in\text{spt}(\mathfrak{m})$ and a Borel set $A\subset\mathsf{X}$ with
$0<\mathfrak{m}(A)<\infty$, there exists a Wasserstein geodesic induced by
$\eta\in\mathscr{P}(\operatorname{Geo}(\mathsf{X}))$ connecting $\delta_{x}$
and $\frac{\mathfrak{m}|_{A}}{\mathfrak{m}(A)}$ such that, for every
$t\in[0,1]$,

$\frac{1}{\mathfrak{m}(A)}\mathfrak{m}\geq(e_{t})_{\\#}\Big{(}\tau_{K,N}^{(t)}\big{(}\mathsf{d}(\gamma(0),\gamma(1))\big{)}^{N}\eta(\text{d}\gamma)\Big{)}.$
(16)

######  Remark 2.6.

For our purposes, we will use an equivalent formulation of the inequality
(16), which holds whenever geodesics are unique, cf. [Oht07, Lemma 2.3] for
further details. More precisely, let $x\in\text{spt}(\mathfrak{m})$ and a
Borel set $A\subset\mathsf{X}$ with $0<\mathfrak{m}(A)<\infty$. Assume that
for every $y\in A$, there exists a unique geodesic
$\gamma_{x,y}:[0,1]\to\mathsf{X}$ joining $x$ and $y$. Then, (16) is verified
for the measures $\delta_{x}$ and $\frac{\mathfrak{m}|_{A}}{\mathfrak{m}(A)}$
if and only if

$\mathfrak{m}\big{(}M_{t}(\\{x\\},A^{\prime}))\big{)}\geq\int_{A^{\prime}}\tau_{K,N}^{(t)}(\mathsf{d}(x,y))^{N}\,\mathrm{d}\mathfrak{m}(y),\qquad\text{for
any Borel }A^{\prime}\subset A.$ (17)

The $\mathsf{MCP}(K,N)$ condition is weaker than the $\mathsf{CD}(K,N)$ one,
i.e.

$\mathsf{CD}(K,N)\implies\mathsf{MCP}(K,N),$

for every $K\in\mathbb{R}$ and every $N\in(1,\infty)$. In Theorem 4.26 (for
the case of non-ample geodesics) and Theorem 5.25 (cf. Theorem 1.8) in the
Heisenberg group, equipped with singular norms, we contradict the
$\mathsf{MCP}(K,N)$ condition. More precisely, we find a counterexample to
(17).

### 2.2 Strictly and strongly convex norms on $\mathbb{R}^{k}$

We say that a function $f:\mathbb{R}^{k}\to\mathbb{R}$ is _strictly convex_
if, for any $x,y\in\mathbb{R}^{k}$ such that $x\neq y$,

$f(tx+(1-t)y)<tf(x)+(1-t)f(y),\qquad\forall\,t\in(0,1).$ (18)

Furthermore, let ${\|\\!\cdot\\!\|}:\mathbb{R}^{k}\to\mathbb{R}_{\geq 0}$ be a
norm, then we say that $f$ is _strongly convex_ with respect to
${\|\\!\cdot\\!\|}$ if there exists $\alpha>0$ such that, for every
$x,y\in\mathbb{R}^{k}$,

$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-\frac{\alpha}{2}t(1-t)\left\lVert
x-y\right\rVert^{2},\qquad\forall\,t\in[0,1].$ (19)

######  Definition 2.7 (Strictly and strongly convex norm).

Let ${\|\\!\cdot\\!\|}:\mathbb{R}^{k}\to\mathbb{R}_{\geq 0}$ be a (symmetric)
norm on $\mathbb{R}^{k}$. We say that ${\|\\!\cdot\\!\|}$ is _strictly convex_
if the function $f:\mathbb{R}^{k}\to\mathbb{R}$ defined by $f(x):=\left\lVert
x\right\rVert^{2}$ is strictly convex. Similarly, we say that
${\|\\!\cdot\\!\|}$ is _strongly convex_ if $f$ is strongly convex with
respect to ${\|\\!\cdot\\!\|}$.

Note that, according to [Cio90, Prop. 1.6], ${\|\\!\cdot\\!\|}$ is strictly
convex if and only if the associated unit ball
$B^{\left\lVert\cdot\right\rVert}_{1}(0)$ is a strictly convex set. Whereas,
[Cio90, Prop. 2.11] implies that ${\|\\!\cdot\\!\|}$ is strongly convex if and
only if the associated unit ball $B^{\left\lVert\cdot\right\rVert}_{1}(0)$ is
a uniformly convex set (see also [KdP21, Thm. 4.1(f)]). We recall below a
well-known result on the relation between a norm ${\|\\!\cdot\\!\|}$ and its
dual norm ${\|\\!\cdot\\!\|}_{*}$, cf. [AP95, Prop. 2.6] or [KdP21, Def. 3.1].

######  Proposition 2.8.

Let ${\|\\!\cdot\\!\|}$ be a norm on $\mathbb{R}^{k}$, and let
${\|\\!\cdot\\!\|}_{*}$ be its dual norm, then:

  * (i)

${\|\\!\cdot\\!\|}$ is a strictly convex norm if and only if
${\|\\!\cdot\\!\|}_{*}$ is a $C^{1}$ norm, i.e. ${\|\\!\cdot\\!\|}_{*}\in
C^{1}(\mathbb{R}^{k}\setminus\\{0\\})$;

  * (ii)

${\|\\!\cdot\\!\|}$ is a strongly convex norm if and only if
${\|\\!\cdot\\!\|}_{*}$ is a $C^{1,1}$ norm, i.e. ${\|\\!\cdot\\!\|}_{*}\in
C^{1,1}(\mathbb{R}^{k}\setminus\\{0\\})$.

We conclude this section by providing an explicit description of the dual
element of $v\in(\mathbb{R}^{k},{\|\\!\cdot\\!\|})$, when ${\|\\!\cdot\\!\|}$
is strictly convex and $C^{1}$. Recall that its dual vector
$v^{*}\in(\mathbb{R}^{k},{\|\\!\cdot\\!\|})^{*}$ is uniquely characterized by

$\left\lVert v^{*}\right\rVert_{*}=\left\lVert
v\right\rVert\qquad\text{and}\qquad\langle v^{*},v\rangle=\left\lVert
v\right\rVert^{2},$ (20)

where $\langle\cdot,\cdot\rangle$ is the dual coupling.

######  Lemma 2.9.

Let $(\mathbb{R}^{k},{\|\\!\cdot\\!\|})$ be a normed space and assume
${\|\\!\cdot\\!\|}:\mathbb{R}^{k}\to\mathbb{R}_{+}$ is a strictly convex
$C^{1}$ norm. Then, for every non-zero vector $v\in\mathbb{R}^{k}$, it holds
that

$v^{*}=\left\lVert v\right\rVert\cdot d_{v}{\|\\!\cdot\\!\|}.$ (21)

###### Proof.

Set
$\lambda:=d_{v}{\|\\!\cdot\\!\|}\in(\mathbb{R}^{k},{\|\\!\cdot\\!\|})^{*}$,
where we recall that

$d_{v}{\|\\!\cdot\\!\|}(u):=\lim_{t\to 0}\frac{\left\lVert
v+tu\right\rVert-\left\lVert
u\right\rVert}{t},\qquad\forall\,u,v\in\mathbb{R}^{k}.$ (22)

Then, on the one hand, it holds that

$\langle\lambda,v\rangle=d_{v}{\|\\!\cdot\\!\|}(v)=\lim_{t\to
0}\frac{\left\lVert v+tv\right\rVert-\left\lVert v\right\rVert}{t}=\left\lVert
v\right\rVert.$ (23)

On the other hand, we have

$\left\lVert\lambda\right\rVert_{*}=\sup_{u\in
B_{1}}\langle\lambda,u\rangle=\sup_{u\in
B_{1}}d_{v}{\|\\!\cdot\\!\|}(u)=\sup_{u\in B_{1}}\lim_{t\to
0}\frac{1}{t}\big{(}\left\lVert v+tu\right\rVert-\left\lVert
v\right\rVert\big{)}\leq\sup_{u\in B_{1}}\left\lVert u\right\rVert=1,$

where $B_{1}:=B_{1}^{\|\cdot\|}(0)\subset\mathbb{R}^{k}$ is the ball of radius
$1$ and centered at $0$, with respect to the norm ${\|\\!\cdot\\!\|}$. The
converse inequality can be obtained by taking $u=\frac{v}{\left\lVert
v\right\rVert}$. Finally, the conclusion follows by homogeneity of the dual
norm. ∎

### 2.3 Sub-Finsler structures

Let $M$ be a smooth manifold of dimension $n$ and let $k\in\mathbb{N}$. A
_sub-Finsler structure_ on $M$ is a couple $(\xi,{\|\\!\cdot\\!\|})$ where
${\|\\!\cdot\\!\|}:\mathbb{R}^{k}\to\mathbb{R}_{+}$ is a norm on
$\mathbb{R}^{k}$ and $\xi:M\times\mathbb{R}^{k}\rightarrow TM$ is a morphism
of vector bundles such that:

  1. (i)

each fiber of the (trivial) bundle $M\times\mathbb{R}^{k}$ is equipped with
the norm ${\|\\!\cdot\\!\|}$;

  2. (ii)

The set of horizontal vector fields, defined as

$\mathcal{D}:=\big{\\{}\xi\circ\sigma\,:\,\sigma\in\Gamma(M\times\mathbb{R}^{k})\big{\\}}\subset\Gamma(TM),$
(24)

is a _bracket-generating_ family of vector fields (or it satisfies the
Hörmander condtion), namely setting

${\rm
Lie}_{q}(\mathcal{D}):=\big{\\{}X(q)\,:\,X\in\text{span}\\{[X_{1},\ldots,[X_{j-1},X_{j}]]\,:\,X_{i}\in\mathcal{D},j\in\mathbb{N}\\}\big{\\}},\qquad\forall\,q\in
M,$ (25)

we assume that ${\rm Lie}_{q}(\mathcal{D})=T_{q}M$, for every $q\in M$.

######  Definition 2.10 (Smooth sub-Finsler manifold).

Let $(\xi,{\|\\!\cdot\\!\|})$ be a sub-Finsler structure on $M$. We say that
$M$ is a _smooth sub-Finsler manifold_ , if the norm ${\|\\!\cdot\\!\|}$ is
strongly convex and smooth, i.e. ${\|\\!\cdot\\!\|}\in
C^{\infty}(\mathbb{R}^{k}\setminus\\{0\\})$.

######  Remark 2.11.

Although this definition is not completely general in sub-Finsler context,
since it does not allow the norm to vary on the fiber of
$M\times\mathbb{R}^{k}$, it includes sub-Riemannian geometry (where
${\|\\!\cdot\\!\|}$ is induced by a scalar product), as every sub-Riemannian
structure is equivalent to a free one, cf. [ABB20, Sec. 3.1.4].

At every point $q\in M$ we define the _distribution_ at $q$ as

$\mathcal{D}_{q}:=\big{\\{}\xi(q,w)\,:\,w\in\mathbb{R}^{k}\big{\\}}=\big{\\{}X(q)\,:\,X\in\mathcal{D}\big{\\}}\subset
T_{q}M.$ (26)

This is a vector subspace of $T_{q}M$ whose dimension is called _rank_ (of the
distribution) and denoted by $r(q):=\dim\mathcal{D}_{q}\leq n$. Moreover, the
distribution is described by a family of horizontal vector fields. Indeed,
letting $\\{e_{i}\\}_{i=1,\dots,k}$ be the standard basis of $\mathbb{R}^{k}$,
the _generating frame_ is the family $\\{X_{i}\\}_{i=1,\dots,k}$, where

$X_{i}(q):=\xi(q,e_{i})\qquad\forall\,q\in M,\quad\text{for }i=1,\dots,k.$

Then, according to (26),
$\mathcal{D}_{q}=\text{span}\\{X_{1}(q),\dots,X_{k}(q)\\}$. On the
distribution we define the _induced norm_ as

$\left\lVert v\right\rVert_{q}:=\inf\big{\\{}\left\lVert
w\right\rVert\,:\,v=\xi(q,w)\big{\\}}\qquad\text{for every
}v\in\mathcal{D}_{q}.$

Since the infimum is actually a minimum, the function ${\|\\!\cdot\\!\|}_{q}$
is a norm on $\mathcal{D}_{q}$, so that
$(\mathcal{D}_{q},{\|\\!\cdot\\!\|}_{q})$ is a normed space. Moreover, the
norm depends smoothly on the base point $q\in M$. A curve $\gamma:[0,1]\to M$
is _admissible_ if its velocity $\dot{\gamma}(t)$ exists almost everywhere and
there exists a function $u=(u_{1},\ldots,u_{k})\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$ such that

$\dot{\gamma}(t)=\sum_{i=1}^{k}u_{i}(t)X_{i}(\gamma(t)),\qquad\text{for a.e.
}t\in[0,1].$ (27)

The function $u$ is called _control_. Furthermore, given an admissible curve
$\gamma$, there exists
$\bar{u}=(\bar{u}_{1},\dots,\bar{u}_{k}):[0,1]\to\mathbb{R}^{k}$ such that

$\dot{\gamma}(t)=\sum_{i=1}^{k}\bar{u}_{i}(t)X_{i}(\gamma(t)),\qquad\text{and}\qquad\left\lVert\dot{\gamma}(t)\right\rVert_{\gamma(t)}=\left\lVert\bar{u}(t)\right\rVert,\qquad\text{for
a.e. }t\in[0,1].$ (28)

The function $\bar{u}$ is called _minimal control_ , and it belongs to
$L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$, cf. [ABB20,
Lem. 3.12]. We define the _length_ of an admissible curve:

$\ell(\gamma):=\int_{0}^{1}\left\lVert\dot{\gamma}(t)\right\rVert_{\gamma(t)}\,\mathrm{d}t\in[0,\infty).$
(29)

We can rewrite the length of a curve as the $L^{1}$-norm of the associated
minimal control, indeed by (28),

$\ell(\gamma)=\int_{0}^{1}\left\lVert\bar{u}(t)\right\rVert\,\mathrm{d}t=\left\lVert\bar{u}\right\rVert_{L^{1}([0,1];(\mathbb{R}^{k},\|\cdot\|))}.$
(30)

For every couple of points $q_{0},q_{1}\in M$, define the _sub-Finsler
distance_ between them as

$\mathsf{d}_{SF}(q_{0},q_{1})=\inf\left\\{\ell(\gamma)\,:\,\gamma\text{
admissible, }\gamma(0)=q_{0}\text{ and }\gamma(1)=q_{1}\right\\}.$

Since every norm on $\mathbb{R}^{k}$ is equivalent to the standard scalar
product on $\mathbb{R}^{k}$, it follows that the sub-Riemannian structure on
$M$ given by $(\xi,\langle\cdot,\cdot\rangle)$ induces an equivalent distance.
Namely, denoting by $\mathsf{d}_{SR}$ the induced sub-Riemannian distance,
there exist constants $C>c>0$ such that

$c\,\mathsf{d}_{SR}\leq\mathsf{d}_{SF}\leq C\mathsf{d}_{SR},\qquad\text{on
}M\times M.$ (31)

Thus, as a consequence of the classical Chow–Rashevskii Theorem in sub-
Riemannian geometry, we obtain the following.

######  Proposition 2.12 (Chow–Rashevskii).

Let $M$ be a sub-Finsler manifold. The sub-Finsler distance is finite,
continuous on $M\times M$ and the induced topology is the manifold one.

From this proposition, we get that $(M,\mathsf{d}_{SF})$ is a locally compact
metric space. The local existence of minimizers of the length functional can
be obtained as in the sub-Riemannian setting, in particular, one can repeat
the proof of [ABB20, Thm. 3.43]. Finally, if $(M,\mathsf{d}_{SF})$ is
complete, then it is also a geodesic metric space.

## 3 The geometry of smooth sub-Finsler manifolds

### 3.1 The energy functional and the optimal control problem

Let $\gamma:[0,1]\rightarrow M$ be an admissible curve. Then, we define the
_energy_ of $\gamma$ as

$J(\gamma)=\frac{1}{2}\int_{0}^{1}\left\lVert\dot{\gamma}(t)\right\rVert^{2}_{\gamma(t)}\,\mathrm{d}t.$
(32)

By definition of admissible curve, $J(\gamma)<+\infty$. In addition, a
standard argument shows that $\gamma:[0,1]\rightarrow M$ is a minimum for the
energy functional if and only if it is a minimum of the length functional with
constant speed.

######  Remark 3.1.

The minimum of $J$ is not invariant under reparametrization of $\gamma$, so
one needs to fix the interval where the curve is defined. Here and below, we
choose $[0,1]$.

The problem of finding geodesics between two points $q_{0},q_{1}\in M$ can be
formulated using the energy functional as the following costrained
minimization problem:

$\begin{cases}\gamma:[0,1]\to M,\quad\text{admissible},\\\
\gamma(0)=q_{0}\text{ and }\gamma(1)=q_{1},\\\
J(\gamma)=\displaystyle\frac{1}{2}\int_{0}^{1}\left\lVert\dot{\gamma}(t)\right\rVert^{2}_{\gamma(t)}\,\mathrm{d}t\to\min.\end{cases}$
(P)

The problem (P) can be recasted as an optimal control problem. First of all, a
curve is admissible if and only if there exists a control in
$L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$ satisfying (27).
Second of all, we can consider the energy as a functional on the space of
controls. Indeed, as in (30), given an admissible curve $\gamma:[0,1]\to M$,
we let $\bar{u}\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$ be its minimal
control, as in (28). Then, we have

$J(\gamma)=\frac{1}{2}\int_{0}^{1}\left\lVert\bar{u}(t)\right\rVert^{2}\,\mathrm{d}t=\frac{1}{2}\left\lVert\bar{u}\right\rVert^{2}_{L^{2}([0,1];(\mathbb{R}^{k},\|\cdot\|))}.$
(33)

Hence, we regard the energy as a functional on
$L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$, namely

$J:L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}\to\mathbb{R}_{+};\qquad
J(u):=\frac{1}{2}\left\lVert
u\right\rVert^{2}_{L^{2}([0,1];(\mathbb{R}^{k},\|\cdot\|))},$ (34)

and we look for a constrained minimum of it. Thus, the problem (P) becomes:

$\begin{cases}\dot{\gamma}(t)=\displaystyle\sum_{i=1}^{k}u_{i}(t)X_{i}(\gamma(t)),\\\
\gamma(0)=q_{0}\text{ and }\gamma(1)=q_{1},\\\
J(u)=\displaystyle\frac{1}{2}\int_{0}^{1}\left\lVert
u(t)\right\rVert^{2}\,\mathrm{d}t\to\min.\end{cases}$ (P′)

Note that, by (33), the solutions of (P) and (P′) coincide. An application of
Pontryagin Maximum Principle (see [AS04, Thm. 12.10]) yields necessary
conditions for optimality. For every $u\in\mathbb{R}^{k}$ and
$\nu\in\mathbb{R}$, introduce the following Hamiltonian:

$h_{u}^{\nu}(\lambda):=\langle\lambda,\xi(\pi(\lambda),u)\rangle+\frac{\nu}{2}\left\lVert
u\right\rVert^{2},\qquad\forall\lambda\in T^{*}M.$ (35)

Recall that for $h\in C^{1}(T^{*}M)$, its Hamiltonian vector field
$\vec{h}\in{\rm Vec}(T^{*}M)$ is defined as the unique vector field in
$T^{*}M$ satisfying

$d_{\lambda}h=\sigma(\cdot,\vec{h}(\lambda)),\qquad\forall\,\lambda\in
T^{*}M,$ (36)

where $\sigma$ is the canonical symplectic form on $T^{*}M$.

######  Theorem 3.2 (Pontryagin Maximum Principle).

Let $M$ be a sub-Finsler manifold and let $(\gamma,\bar{u})$ be a solution of
(P′). Then, there exists $(\nu,\lambda_{t})\neq 0$, where $\nu\in\mathbb{R}$
and $\lambda_{t}\in T^{*}_{\gamma(t)}M$ for every $t\in[0,1]$, such that

$\begin{cases}\displaystyle\dot{\lambda}_{t}=\vec{h}_{\bar{u}(t)}^{\nu}(\lambda_{t})&\text{for
a.e. }t\in[0,1],\\\ \displaystyle
h_{\bar{u}(t)}^{\nu}(\lambda_{t})=\max_{v\in\mathbb{R}^{k}}h_{v}^{\nu}(\lambda_{t})&\text{for
a.e. }t\in[0,1],\\\ \nu\leq 0.&\end{cases}$ (H)

######  Definition 3.3.

If $\nu<0$ in (H), $(\lambda_{t})_{t\in[0,1]}$ is called _normal extremal_. If
$\nu=0$ in (H), $(\lambda_{t})_{t\in[0,1]}$ is called _abnormal extremal_.

######  Remark 3.4.

By homogeneity of the Hamiltonian system, if $\nu\neq 0$ in (H) we can fix
$\nu=-1$.

### 3.2 Characterization of extremals and the exponential map

In this section, we recall some characterizations of normal and abnormal
extremal, which are well-known in sub-Riemannian geometry. We include the
proofs in our case, for the sake of completeness.

Recall that the annihilator $\operatorname{Ann}(\mathcal{D})\subset T^{*}M$ is
defined by

$\operatorname{Ann}(\mathcal{D})_{q}:=\\{\lambda\in
T^{*}_{q}M\,:\,\langle\lambda,w\rangle=0,\,\forall\,w\in\mathcal{D}_{q}\\},\qquad\forall\,q\in
M.$ (37)

######  Lemma 3.5.

Let $M$ be a sub-Finsler manifold. Let $(\gamma,\bar{u})$ be a non-trivial
solution to (P′) and let $(\lambda_{t})_{t\in[0,1]}$ be its lift. Then,
$(\lambda_{t})_{t\in[0,1]}$ is an abnormal extremal if and only if
$\lambda_{t}\neq 0$ and
$\lambda_{t}\in\operatorname{Ann}(\mathcal{D})_{\gamma(t)}$ for every
$t\in[0,1]$, where $\gamma(t):=\pi(\lambda_{t})$.

###### Proof.

The claim is an easy consequence of the maximization property of the
Hamiltonian along the dynamic. More precisely, by (H), we have

$h_{\bar{u}(t)}^{\nu}(\lambda_{t})=\max_{v\in\mathbb{R}^{k}}h_{v}^{\nu}(\lambda_{t}),\qquad\text{for
a.e. }t\in[0,1],$ (38)

where the function $h_{u}^{\nu}$ is defined in (35). Assume that $\nu=0$, then
(38) reads as

$\langle\lambda_{t},\xi(\gamma(t),\bar{u}(t))\rangle=\max_{v\in\mathbb{R}^{k}}\,\langle\lambda_{t},\xi(\gamma(t),v)\rangle,\qquad\text{for
a.e. }t\in[0,1],$ (39)

with $\lambda_{t}\neq 0$ for every $t\in[0,1]$. Now, since $\xi$ is linear in
the controls, the right-hand side is $+\infty$ unless
$\lambda_{t}\in\operatorname{Ann}(\mathcal{D})_{\gamma(t)}$ for a.e.
$t\in[0,1]$. By continuity of $t\mapsto\lambda_{t}$, this is true for every
$t\in[0,1]$. Conversely, assume that
$\lambda_{t}\in\operatorname{Ann}(\mathcal{D})_{\gamma(t)}$, then the
maximization condition (38) becomes

$\frac{\nu}{2}\|\bar{u}(t)\|^{2}=\max_{v\in\mathbb{R}^{k}}\frac{\nu}{2}\|v\|^{2}.$
(40)

Since $\nu\leq 0$, we may distinguish two cases, either $\nu=0$ and the
extremal is abnormal, or $\nu=-1$ and the extremal is normal. In the second
case, the optimal control must be $0$, so that $\dot{\lambda}_{t}=0$ and the
extremal is constant and constantly equal to $\lambda_{0}$. Since we are
assuming $(\lambda_{t})_{t\in[0,1]}$ to be non-constant the latter can not
happen. ∎

The _sub-Finsler (or maximized) Hamiltonian_ is defined as

$H(\lambda):=\max_{u\in\mathbb{R}^{k}}h_{u}^{-1}(\lambda_{t})=\max_{u\in\mathbb{R}^{k}}\Bigg{(}\sum_{i=1}^{k}\langle\lambda,u_{i}X_{i}(\pi(\lambda))\rangle-\frac{\left\lVert
u\right\rVert^{2}}{2}\Bigg{)}$ (41)

The sub-Finsler Hamiltonian can be explicitly characterized in terms of the
dual norm ${\|\\!\cdot\\!\|}_{*}$, making use of Lemma 2.9.

######  Lemma 3.6.

Let $M$ be a smooth sub-Finsler manifold. Given $\lambda\in T^{*}M$, define
$\hat{\lambda}=(\hat{\lambda}_{i})_{i=1,\ldots,k}$ where
$\hat{\lambda}_{i}:=\langle\lambda,X_{i}(\pi(\lambda))\rangle$ for every
$i=1,\ldots,k$. Then, $H^{-1}(0)=\operatorname{Ann}(\mathcal{D})$ and

$H(\lambda)=\frac{1}{2}\big{\|}\hat{\lambda}\big{\|}^{2}_{*}\,,\qquad\forall\lambda\in
T^{*}M\setminus\operatorname{Ann}(\mathcal{D}).$ (42)

where ${\|\\!\cdot\\!\|}_{*}$ is the dual norm to ${\|\\!\cdot\\!\|}$ in
$\mathbb{R}^{k}$. Moreover, $H\in
C^{\infty}(T^{*}M\setminus\operatorname{Ann}(\mathcal{D}))\cap C^{1}(T^{*}M)$.

###### Proof.

Let $q\in M$. Assume that $\lambda\in
T_{q}^{*}M\setminus\operatorname{Ann}(\mathcal{D})_{q}$ and set

$F(u):=\langle\hat{\lambda},u\rangle-\frac{\left\lVert
u\right\rVert^{2}}{2},\qquad\forall u\in\mathbb{R}^{k}.$ (43)

Since ${\|\\!\cdot\\!\|}$ is smooth, its square is a $C^{1}$-function, thus
$F\in C^{1}(\mathbb{R}^{k})$. Moreover, by homogeneity of the norm,
$F(u)\to-\infty$ as $\left\lVert u\right\rVert\to\infty$, hence $F$ admits a
maximum. We compute its differential:

$d_{u}F=\hat{\lambda}-\left\lVert u\right\rVert\cdot
d_{u}{\|\\!\cdot\\!\|}=\hat{\lambda}-u^{*},$ (44)

according to Lemma 2.9. Therefore, $F$ has a unique critical point (which is
also the unique point of maximum) given by $u=u^{**}=\hat{\lambda}^{*}$.
Finally, using also (20), this implies that

$H(\lambda)=\max_{u\in\mathbb{R}^{k}}F(u)=F(\hat{\lambda}^{*})=\langle\hat{\lambda},\hat{\lambda}^{*}\rangle-\frac{1}{2}\big{\|}\hat{\lambda}\big{\|}_{*}^{2}=\frac{1}{2}\big{\|}\hat{\lambda}\big{\|}_{*}^{2}.$
(45)

To conclude, observe that if $\lambda\in\operatorname{Ann}(\mathcal{D})_{q}$,
then $\hat{\lambda}=0$ and

$H(\lambda)=\max_{u\in\mathbb{R}^{k}}\left(-\frac{\left\lVert
u\right\rVert^{2}}{2}\right)=0.$ (46)

Conversely, if $H(\lambda)=0$ we must have
$\lambda\in\operatorname{Ann}(\mathcal{D})$. Indeed, if this is not the case,
$\hat{\lambda}\neq 0$ and hence $\big{\|}\hat{\lambda}\big{\|}_{*}\neq 0$,
giving a contradiction. This proves that
$H^{-1}(0)=\operatorname{Ann}(\mathcal{D})$.

Finally, we prove the regularity of $H$. Note that ${\|\\!\cdot\\!\|}_{*}$ is
a smooth norm itself. Indeed, as ${\|\\!\cdot\\!\|}$ is smooth and strongly
convex, the dual map of Lemma 2.9, which is

$v^{*}=\left\lVert v\right\rVert
d_{v}{\|\\!\cdot\\!\|}=\frac{1}{2}d_{v}\big{(}{\|\\!\cdot\\!\|}^{2}\big{)}=:N(v),$
(47)

is smooth on $\\{v\neq 0\\}$, invertible and with invertible differential on
$\\{v\neq 0\\}$. Thus, by the inverse function theorem, $N^{-1}\in
C^{\infty}(\mathbb{R}^{k}\setminus\\{0\\})$. But now the dual norm satisfies
(20), hence

$\|\hat{\lambda}\|_{*}=\|N^{-1}(\hat{\lambda})\|,$ (48)

and the claim follows. Therefore, we deduce that $H\in
C^{\infty}(T^{*}M\setminus H^{-1}(0))\cap
C^{1}(T^{*}M)=C^{\infty}(T^{*}M\setminus\operatorname{Ann}(\mathcal{D}))\cap
C^{1}(T^{*}M)$. ∎

######  Corollary 3.7.

Let $(\lambda_{t})_{t\in[0,1]}$ be a normal extremal for the problem (H), then
the associated control is given by

$\bar{u}(t)=\hat{\lambda}_{t}^{*},\qquad\text{for a.e. }t\in[0,1].$ (49)

###### Proof.

This is again a consequence of the maximality condition in (H), together with
the characterization of the sub-Finsler Hamiltonian. In particular, we must
have

$h^{-1}_{\bar{u}(t)}(\lambda_{t})=\max_{u\in\mathbb{R}^{k}}h^{-1}_{u}(\lambda_{t})=H(\lambda_{t}),\qquad\forall\,t\in[0,1],$
(50)

From this and (44), we deduce that the control associated with
$(\lambda_{t})_{t\in[0,1]}$ satisfies the identity (49). ∎

The next result relates the system (H) with the Hamiltonian system associated
with the sub-Finsler Hamiltonian (41). A similar statement can be found in
[AS04, Prop. 12.3]. The main difference is the regularity of the Hamiltonian
function, which in the classical statement is assumed to be smooth outside the
zero section.

######  Proposition 3.8.

Let $M$ be a smooth sub-Finsler manifold. Let $H\in
C^{\infty}(T^{*}M\setminus\operatorname{Ann}(\mathcal{D}))\cap C^{1}(T^{*}M)$
be the sub-Finsler Hamiltonian defined in (41). If $(\lambda_{t})_{t\in[0,1]}$
is a normal extremal, and $\lambda_{0}\in\operatorname{Ann}(\mathcal{D})$,
then $\lambda_{t}\equiv\lambda_{0}$. If $\lambda_{0}\in
T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$, then

$\dot{\lambda}_{t}=\vec{H}(\lambda_{t}).$ (51)

Conversely, if $(\lambda_{t})_{t\in[0,1]}$ is a solution of (51) with initial
condition $\lambda_{0}\in T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$,
then there exists $\bar{u}\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$ such that
$(\lambda_{t})_{t\in[0,1]}$ is a normal extremal with control $\bar{u}$ (i.e.
the pair $(-1,(\lambda_{t}))$ is a solution of (H)).

###### Proof.

If $(\lambda_{t})_{t\in[0,1]}$ is a normal extremal, there exists an optimal
control $\bar{u}$ such that the pair $(-1,(\lambda_{t}))$ is a solution to
(H). Now, if the initial covector
$\lambda_{0}\in\operatorname{Ann}(\mathcal{D})$, then
$\lambda_{t}\in\operatorname{Ann}(\mathcal{D})$ for all $t\in[0,1]$, as the
sub-Finsler Hamiltonian is constant along the motion and $H(\lambda_{0})=0$.
Using Corollary 3.7, this implies that the control $\bar{u}\equiv 0$ and that
$\lambda_{t}\equiv\lambda_{0}$ as claimed. If $\lambda_{0}\in
T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$, we follow the blueprint of
[AS04, Prop. 12.3]. By the definition of sub-Finsler Hamiltonian (41), we have

$H(\lambda)\geq h_{\bar{u}(t)}(\lambda),\qquad\forall\,\lambda\in
T^{*}M,\,t\in[0,1],$ (52)

with equality along the dynamic $t\mapsto\lambda_{t}$. This means that the
function $T^{*}M\ni\lambda\mapsto H(\lambda)-h_{\bar{u}(t)}(\lambda)$ has a
maximum at $\lambda_{t}$. Therefore, using that $H\in C^{1}(T^{*}M)$, we
deduce that

$d_{\lambda_{t}}H=d_{\lambda_{t}}h_{\bar{u}(t)},\qquad\forall\,t\in[0,1].$
(53)

Such an equality immediately implies that the Hamiltonian vector fields are
equal along the dynamic, namely

$\vec{H}(\lambda_{t})=\vec{h}_{\bar{u}(t)}(\lambda_{t}),\qquad\forall\,t\in[0,1].$
(54)

For the converse implication, recall that the Hamiltonian is constant along
the motion. So, if $(\lambda_{t})_{t\in[0,1]}$ is a solution to (51) with
initial condition $\lambda_{0}\in
T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$, then
$H(\lambda_{0})=H(\lambda_{t})$ and $\lambda_{t}\in
T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$, for every $t\in[0,1]$. Since
$H$ is smooth outside the annihilator bundle of $\mathcal{D}$, we deduce that
$(\lambda_{t})_{t\in[0,1]}$ is uniquely determined by $\lambda_{0}$ and,
repeating verbatim the argument of [AS04, Prop. 12.3], we conclude the proof.
∎

Fix $t\in\mathbb{R}$ and consider the _(reduced) flow of_ $\vec{H}$ on
$T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$:

$e_{\mathrm{r}}^{t\vec{H}}:\mathscr{A}_{t}\to
T^{*}M\setminus\operatorname{Ann}(\mathcal{D}),$ (55)

where $\mathscr{A}_{t}\subset T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$
is the set of covectors such that the associated maximal solution
$(\lambda_{s})_{s\in I}$, with $I\subset\mathbb{R}$ such that $0\in I$, is
defined up to time $t$. Under the assumption of completeness of
$(M,\mathsf{d}_{SF})$, $\vec{H}$ is complete as a vector field on
$T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$ (and thus
$\mathscr{A}_{t}=T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$). We state
below this result without proof as the latter is analogous to the classical
sub-Riemannian proof, cf. [ABB20, Prop. 8.38], in view of Lemma 2.9 and Lemma
3.6.

######  Proposition 3.9.

Let $M$ be a smooth complete sub-Finsler manifold. Then any normal extremal
$t\mapsto\lambda_{t}=e^{t\vec{H}}_{\mathrm{r}}(\lambda_{0})$, with
$\lambda_{0}\in T^{*}M\setminus\operatorname{Ann}(\mathcal{D})$, is extendable
to $\mathbb{R}$.

######  Definition 3.10 (Sub-Finsler exponential map).

Let $(M,\mathsf{d}_{SF})$ be a complete smooth sub-Finsler manifold and let
$q\in M$. Then, the _sub-Finsler exponential map_ at $q$ is defined as

$\exp_{q}(\lambda):=\begin{cases}\pi\circ e^{\vec{H}}_{\rm
r}(\lambda)&\text{if }\lambda\in
T^{*}_{q}M\setminus\operatorname{Ann}(\mathcal{D})_{q},\\\ q&\text{if
}\lambda\in\operatorname{Ann}(\mathcal{D})_{q}.\end{cases}$ (56)

######  Remark 3.11.

The exponential map is smooth in
$T_{q}^{*}M\setminus\operatorname{Ann}(\mathcal{D})_{q}$ and, by homogeneity,
we also have $\exp_{q}\in C^{1}(T^{*}_{q}M)$. However, note that the we do not
have spatial regularity. More precisely, setting $\mathcal{E}:T^{*}M\to M$ to
be $\mathcal{E}(q,\lambda)=\exp_{q}(\lambda)$, then

$\mathcal{E}\in C(T^{*}M)\cap
C^{\infty}(T^{*}M\setminus\operatorname{Ann}(\mathcal{D})),$ (57)

and we can not expect a better spatial regularity as the vector field
$\vec{H}$ is only continuous on the annihilator bundle.

### 3.3 The end-point map

Let $M$ be a sub-Finsler manifold, consider $u\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$ and fix
$t_{0}\in[0,1]$. Define the non-autonomous vector field

$\xi_{u(t)}(q):=\xi(q,u(t))=\sum_{i=1}^{k}u_{i}(t)X_{i}(q),\qquad\forall\,q\in
M,t\in[0,1],$ (58)

and denote by $P^{u}_{t_{0},t}:M\to M$ its flow. This means that, for
$q_{0}\in M$, the curve $t\mapsto P^{u}_{t_{0},t}(q_{0})$ is the unique
maximal solution to the Cauchy problem

$\begin{cases}\displaystyle\dot{\gamma}(t)=\sum_{i=1}^{m}u_{i}(t)X_{i}(\gamma(t)),\\\
\gamma(t_{0})=q_{0}.\end{cases}$

Moreover, we denote by $\gamma_{u}:I\to M$, the trajectory starting at $q_{0}$
and corresponding to $u$, namely $\gamma_{u}(t):=P_{0,t}^{u}(q_{0})$, for
every $t\in I$, which is the maximal interval of definition of $\gamma_{u}$.

######  Definition 3.12 (End-point map).

Let $M$ be a sub-Finsler manifold and let $q_{0}\in M$. We call
$\mathcal{U}_{q_{0}}\subset
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$ the open set of
controls for which the corresponding trajectory $\gamma_{u}$ is defined on the
interval $[0,1]$. We define the _end-point map_ based at $q_{0}$ as

$E_{q_{0}}:\mathcal{U}_{q_{0}}\to M,\qquad E_{q_{0}}(u)=\gamma_{u}(1).$

######  Lemma 3.13 (Differential of the end-point map, [ABB20, Prop. 8.5]).

Let $M$ be a sub-Finsler manifold. The end-point map is smooth on
$\mathcal{U}_{q_{0}}$. Moreover, for every $u\in\mathcal{U}_{q_{0}}$ the
differential
$d_{u}E_{q_{0}}:L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}\rightarrow
T_{E_{q_{0}}(u)}M$ has the expression

$d_{u}E_{q_{0}}(v)=\int_{0}^{1}\left(P_{t,1}^{u}\right)_{*}\xi_{v(t)}(E_{q_{0}}(u))\,\mathrm{d}t,\qquad\text{for
every }v\in L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}.$

Using the explicit expression for the differential of the end-point map we
deduce the following characterization for normal and abnormal extremals in
sub-Finsler geometry, see [ABB20, Prop. 8.9] for the analogous sub-Riemannian
result.

######  Proposition 3.14.

Let $M$ be a smooth sub-Finsler manifold and let $(\gamma,u)$ be a non-trivial
solution to (P′). Then, there exists $\lambda_{1}\in T_{q_{1}}^{*}M$, where
$q_{1}=E_{q_{0}}(u)$, such that the curve $(\lambda_{t})_{t\in[0,1]}$, with

$\lambda_{t}:=(P_{t,1}^{u})^{*}\lambda_{1}\in
T^{*}_{\gamma(t)}M,\qquad\forall\,t\in[0,1],$ (59)

is a solution to (H). Moreover, one of the following conditions is satisfied:

  * (i)

$(\lambda_{t})_{t\in[0,1]}$ is a normal extremal if and only if $u$ satisfies

$\langle\lambda_{1},d_{u}E_{q_{0}}(v)\rangle=\langle
u^{*},v\rangle\qquad\text{for every }v\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)};$ (60)

  * (ii)

$(\lambda_{t})_{t\in[0,1]}$ is an abnormal extremal if and only if $u$
satisfies

$\langle\lambda_{1},d_{u}E_{q_{0}}(v)\rangle=0\qquad\text{for every }v\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}.$ (61)

###### Proof.

Firstly, (59) is a well-known consequence of the Pontryagin maximum principle.
We prove (i), the proof of (ii) is analogous. For every $v\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$, using Lemma 3.13
we deduce that

$\begin{split}\langle\lambda_{1},d_{u}E_{q_{0}}(v)\rangle&=\int_{0}^{1}\langle\lambda_{1},(P^{u}_{t,1})_{*}\xi_{v(t)}(E_{q_{0}}(u))\rangle\,\mathrm{d}t=\int_{0}^{1}\langle(P^{u}_{t,1})^{*}\lambda_{1},\xi_{v(t)}(\gamma(t))\rangle\,\mathrm{d}t\\\
&=\int_{0}^{1}\langle\lambda_{t},\xi_{v(t)}(\gamma(t))\rangle\,\mathrm{d}t=\int_{0}^{1}\sum_{i=1}^{k}\langle\lambda_{t},\xi_{i}(\gamma(t))\rangle
v_{i}(t)\,\mathrm{d}t.\end{split}$ (62)

Assume $(\lambda_{t})_{t\in[0,1]}$ is a normal extremal, then, by Corollary
3.7, the associated optimal control $u$ satisfies
$\langle\lambda_{t},\xi_{i}(\gamma(t))\rangle=u^{*}_{i}(t)$ for $i=1,\dots,k$.
Therefore, using (62) we deduce that for every $v\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$ it holds

$\langle\lambda_{1},d_{u}E_{q_{0}}(v)\rangle=\int_{0}^{1}\sum_{i=1}^{k}\langle\lambda_{t},\xi_{i}(\gamma(t))\rangle
v_{i}(t)\,\mathrm{d}t=\int_{0}^{1}\langle
u^{*}(t),v(t)\rangle\,\mathrm{d}t=\langle u^{*},v\rangle.$

This proves (60).

Conversely, assume that the control $u$ satisfies (60). We are going to prove
that $(\lambda_{t})_{t\in[0,1]}$ is a normal extremal. Using (60) and (62) we
deduce that for every $v\in
L^{2}\big{(}[0,1];(\mathbb{R}^{k},{\|\\!\cdot\\!\|})\big{)}$

$\langle
u^{*},v\rangle_{L^{2}}=\langle\lambda_{1},d_{u}E_{q_{0}}(v)\rangle=\int_{0}^{1}\sum_{i=1}^{k}\langle\lambda_{t},\xi_{i}(\gamma(t))\rangle
v_{i}(t)\,\mathrm{d}t,$

As a consequence, since $v$ is arbitrary, we conclude that
$\langle\lambda_{t},\xi_{i}(\gamma(t))\rangle=u^{*}_{i}(t)$ for $i=1,\dots,k$.
∎

## 4 Failure of the $\mathsf{CD}(K,N)$ condition in smooth sub-Finsler
manifolds

In this section, we prove our main result regarding smooth sub-Finsler
manifolds, cf. Theorem 1.5. One of the crucial ingredients for the proof is
the construction of a geodesic, enjoying good regularity properties, cf.
Theorem 4.11.

### 4.1 Construction of a geodesic without abnormal sub-segments

This section is devoted to the construction of a geodesic without abnormal
sub-segments, in smooth sub-Finsler manifolds. The main idea is to choose a
short segment of a normal geodesic that minimizes the distance from a
hypersurface without characteristic points. We recall the definition of
strongly normal geodesic and of geodesic without abnormal sub-segments.

######  Definition 4.1.

Let $M$ be a sub-Finsler manifold and let $\gamma:[0,1]\to M$ be a normal
geodesic. Then, we say that $\gamma$ is

  1. (i)

_left strongly normal_ , if for all $s\in[0,1]$, the restriction
$\gamma|_{[0,s]}$ is not abnormal;

  2. (ii)

_right strongly normal_ , if for all $s\in[0,1]$, the restriction
$\gamma|_{[s,1]}$ is not abnormal;

  3. (iii)

_strongly normal_ , if $\gamma$ is left and right strongly normal.

Finally, we say that $\gamma$ does not admit abnormal sub-segments if any
restriction of $\gamma$ is strongly normal.

Let $\Sigma\subset M$ be a hypersurface and let $\gamma\colon[0,T]\rightarrow
M$ be a horizontal curve, parameterized with constant speed, such that
$\gamma(0)\in\Sigma$, $\gamma(T)=p\in M\setminus\Sigma$. Assume $\gamma$ is a
minimizer for $\mathsf{d}_{SF}(\cdot,\Sigma)$, that is
$\ell(\gamma)=\mathsf{d}_{SF}(p,\Sigma)$. Then, $\gamma$ is a geodesic and any
corresponding normal or abnormal lift, say $\lambda:[0,T]\to T^{*}M$, must
satisfy the transversality conditions, cf. [AS04, Thm 12.13],

$\langle\lambda_{0},w\rangle=0,\qquad\forall\,w\in T_{\gamma(0)}\Sigma.$ (63)

Equivalently, the initial covector $\lambda_{0}$ must belong to the
annihilator bundle $\operatorname{Ann}(\Sigma)$ of $\Sigma$ with fiber
$\operatorname{Ann}(\Sigma)_{q}=\\{\lambda\in
T_{q}^{*}M\mid\langle\lambda,T_{q}\Sigma\rangle=0\\}$, for any $q\in\Sigma$.

######  Remark 4.2.

In the sub-Riemannian setting, the normal exponential map $E$, defined as the
restriction of the exponential map to the annihilator bundle of $\Sigma$,
allows to build (locally) a smooth tubular neighborhood around non-
characteristic points, cf. [FPR20, Prop. 3.1]. This may fail in the sub-
Finsler setting as $E$ is not regular at $\Sigma$, cf. Remark 3.11.
Nonetheless, we are able to deduce a weaker result that is enough for our
construction, see Theorem 4.8.

Recall that $q\in\Sigma$ is a _characteristic point_ , and we write $q\in
C(\Sigma)$, if $\mathcal{D}_{q}\subset T_{q}\Sigma$. As it happens in the sub-
Riemannian case, also in the sub-Finsler setting, minimizers of
$\mathsf{d}_{SF}(\cdot,\Sigma)$ whose initial point is a non-characteristic
point, can not be abnormal geodesics.

######  Lemma 4.3.

Let $M$ be a smooth sub-Finsler manifold. Let $p\in M\setminus\Sigma$ and let
$\gamma:[0,1]\to M$ be a horizontal curve such that

$\gamma(0)\in\Sigma,\quad\gamma(1)=p\quad\text{and}\quad\ell(\gamma)=\mathsf{d}_{SF}(p,\Sigma).$
(64)

Then, $\gamma(0)\in C(\Sigma)$ if and only if $\gamma$ is an abnormal
geodesic.

###### Proof.

The proof is a straightforward adaptation of the analogous result in the sub-
Riemannian setting. We sketch here the argument for completeness. By the
Pontryagin maximum principle, cf. Theorem 3.2, there exists a lift
$\lambda:[0,1]\to T^{*}M$ verifying the system (H) with the additional
condition (63). Using the characterization of Lemma 3.5, $\lambda$ is an
abnormal lift if and only if
$\lambda_{0}\in\operatorname{Ann}(\mathcal{D})_{\gamma(0)}$. The latter,
combined with the transversality condition concludes the proof. ∎

From now on, we assume that $\Sigma$ is the boundary of an open set
$\Omega\subset M$. As our results are local in nature, this assumption is not
necessary, however it makes the presentation easier. Let $\Omega\subset M$ be
a non-characteristic domain in $M$, so that $\partial\Omega$ is compact and
without characteristic points. Then, there exists a never-vanishing smooth
section of $\operatorname{Ann}(\partial\Omega)$, i.e. a smooth map
$\lambda^{+}:\partial\Omega\to\operatorname{Ann}(\partial\Omega)$ such that

$\lambda^{+}(q)\in\operatorname{Ann}_{q}(\partial\Omega)\qquad\text{and}\qquad
2H(\lambda^{+})=1,$ (65)

which is uniquely determined, up to a sign. Define the _normal exponential
map_ as the restriction of the sub-Finsler exponential map to the annihilator
bundle, namely

$E:D\to M,\qquad E(q,\lambda)=\exp_{q}(\lambda),$ (66)

where $D\subset\operatorname{Ann}(\partial\Omega)$ is the largest open sub-
bundle where $E$ is defined. Furthermore, we define the _distance function
from_ $\partial\Omega$ as

$\delta:M\to[0,\infty),\qquad\delta(p):=\mathsf{d}_{SF}(p,\partial\Omega).$
(67)

######  Lemma 4.4.

Let $M$ be a smooth sub-Finsler manifold. There exists $\epsilon>0$ such that
on the sub-bundle

$D_{\epsilon}:=\\{(q,\lambda)\in\operatorname{Ann}(\partial\Omega):E(q,\lambda)\in\Omega\
\text{ and }\ 0<\sqrt{2H(\lambda)}<\epsilon\\}\subset D$ (68)

the map $E|_{D_{\epsilon}}$ is injective and
$E(D_{\epsilon})=\\{0<\delta<\epsilon\\}\cap\Omega$.

###### Proof.

Without loss of generality, we assume that $M$ is complete, so that
$D=\operatorname{Ann}(\partial\Omega)$. We may proceed by contradiction and
assume that there does not exist a choice of $\epsilon>0$ so that
$E|_{D_{\epsilon}}$ is injective. Hence, we can find sequences
$\\{(q_{n},\lambda_{n})\\},\\{(q_{n}^{\prime},\lambda_{n}^{\prime})\\}\subset\operatorname{Ann}(\partial\Omega)$
such that

$(q_{n},\lambda_{n})\neq(q_{n}^{\prime},\lambda_{n}^{\prime})\qquad
E(q_{n},\lambda_{n})=E(q_{n}^{\prime},\lambda_{n}^{\prime}),\qquad\text{and}\qquad
H(\lambda_{n}),\,H(\lambda_{n}^{\prime})\to 0.$ (69)

Note that, as $\partial\Omega$ has no characteristic points, the sub-Finsler
Hamiltonian is a norm on the fibers of $\operatorname{Ann}(\partial\Omega)$.
Therefore, by compactness, $(q_{n},\lambda_{n})\to(q,0)$ and
$(q_{n}^{\prime},\lambda_{n}^{\prime})\to(q^{\prime},0)$, up to subsequences.
Thus, recalling that $E$ is continuous on $D$, passing to the limit in (69),
we get that $E(q,0)=E(q^{\prime},0)$, meaning that $q=q^{\prime}$. As a
consequence
$\lambda_{n},\lambda_{n}^{\prime}\in\operatorname{Ann}(\partial\Omega)_{q}$ so
they are multiple of the section defined in (65), namely

$\lambda_{n}=t_{n}\lambda^{+}(q),\qquad\lambda_{n}^{\prime}=t_{n}^{\prime}\lambda^{+}(q),$
(70)

where $t_{n},t_{n}^{\prime}\to 0$ and their signs agree. Finally, recall that
the length of the normal curve $[0,1]\ni t\mapsto E(q,t\lambda)$ is exactly
$\sqrt{2H(\lambda)}$. This forces $t_{n}=t_{n}^{\prime}$ which is a
contradiction with (69). We are left to prove the last part of the statement.
Fix $\epsilon>0$ so that $E|_{D_{\epsilon}}$ is injective, then
$E(D_{\epsilon})\subset\\{0<\delta<\epsilon\\}\cap\Omega$. For the converse
inclusion, pick $p\in\\{0<\delta<\epsilon\\}\cap\Omega$ and let
$\gamma:[0,1]\to M$ be a geodesic joining $\gamma(0)\in\partial\Omega$ and
$p=\gamma(1)$ such that $\ell(\gamma)=\delta(p)$. Then, $\gamma(0)$ is a non-
characteristic point, therefore $\gamma$ is a normal geodesic, whose lift
satisfies (63), according to Lemma 4.3. Hence, there exists
$0\neq\lambda\in\operatorname{Ann}(\partial\Omega)_{\gamma(0)}$ such that
$\gamma(t)=E(\gamma(0),t\lambda)$ and
$\ell(\gamma)=\sqrt{2H(\lambda)}<\epsilon$. Thus, $(q,\lambda)\in
D_{\epsilon}$ concluding the proof. ∎

We state here a useful lemma regarding the regularity of the distance function
from a boundary. Recall that a function $f:M\to\mathbb{R}$ is said to be
_locally semiconcave_ if, for every $p\in M$, there exist a coordinate chart
$\varphi:U\subset M\to\mathbb{R}^{n}$, with $p\in U$, and a constant
$C\in\mathbb{R}$ such that

$F:\mathbb{R}^{n}\to\mathbb{R};\qquad
F(x):=f\circ\varphi^{-1}(x)-C\frac{|x|^{2}}{2},$ (71)

is concave, where $|\cdot|$ denotes the Euclidean norm.

######  Lemma 4.5.

Let $M$ be a smooth sub-Finsler manifold. Let $\Omega\subset M$ be an open and
bounded subset. Assume that $\partial\Omega$ is smooth and without
characteristic points. Then, the distance function from $\partial\Omega$,
$\delta$ is locally semiconcave in $\Omega$.

###### Proof.

We do not report here a complete proof, since it follows the same arguments of
[ACS18], with the obvious modification for the sub-Finsler case. In
particular, applying Lemma 4.3, we deduce there are no abnormal geodesic
joining points of $\Omega$ to its boundary and realizing $\delta$. Thus, the
proof of [ACS18, Thm. 3.2] shows that $\delta$ is locally Lipschitz in
coordinates, meaning that the function $\delta$ written in coordinates is
Lipschitz with respect to the Euclidean distance. Then, using [ACS18, Thm.
4.1] implication $(3)\Rightarrow(2)$, we conclude. ∎

Since $\delta$ is locally semiconcave, Alexandrov’s theorem ensures that
$\delta$ is differentiable two times $\mathscr{L}^{n}$-a.e. (in coordinates)
and, letting $\mathcal{U}\subset\Omega$ be the set where $\delta$ is
differentiable, the function $d\delta:\mathcal{U}\to T^{*}M$ is differentiable
$\mathscr{L}^{n}$-a.e., cf. [ABS21, Thm. 6.4] for the precise statement of
Alexandrov’s theorem. This observation, combined with Lemma 4.6 below, gives
us an alternative description of geodesics joining $\partial\Omega$ and
differentiablity points of $\delta$ in $\\{0<\delta<\epsilon\\}\cap\Omega$.

######  Lemma 4.6.

Let $M$ be a smooth sub-Finsler manifold. Let $p,q\in M$ be distinct points
and assume there is a function $\phi:M\to\mathbb{R}$ differentiable at $p$ and
such that

$\phi(p)=\frac{1}{2}\mathsf{d}^{2}_{SF}(p,q)\qquad\text{and}\qquad\frac{1}{2}\mathsf{d}^{2}_{SF}(z,q)\geq\phi(z),\quad\forall\,z\in
M.$ (72)

Then, the geodesic joining $p$ and $q$ is unique, has a normal lift and is
given by $\gamma:[0,1]\to M$; $\gamma(t)=\exp_{p}(-td_{p}\phi)$.

###### Proof.

This is a well-known result in sub-Riemannian geometry, cf. [Rif14, Lem.
2.15]. The same proof can be carried out without substantial modifications in
the setting of sub-Finsler manifolds, in light of Proposition 3.14. ∎

######  Corollary 4.7.

Let $M$ be a smooth sub-Finsler manifold and let $\Omega\subset M$ be an open
and bounded subset. Assume that $\partial\Omega$ is smooth and without
characteristic points. Let $p\in\\{0<\delta<\epsilon\\}\cap\Omega$ be a
differentiability point of $\delta$. Then, the unique geodesic
$\gamma:[0,1]\to M$ joining $p$ and $\partial\Omega$ and such that
$\delta(p)=\ell(\gamma)$ is defined by
$\gamma(t)=\exp_{p}\big{(}-\frac{t}{2}d_{p}\delta^{2}\big{)}$.

###### Proof.

Since $p\in\\{0<\delta<\epsilon\\}\cap\Omega$, from Lemma 4.4 we know that the
geodesic joining $p$ and $\partial\Omega$, and realizing $\delta$ is normal
and unique. Let $q\in\partial\Omega$ be its endpoint and define

$\phi:M\to\mathbb{R};\qquad\phi(z):=\frac{1}{2}\delta^{2}(z).$ (73)

Note that $\phi$ is differentiable at the point $p$,
$\phi(p)=\frac{1}{2}\ell(\gamma)^{2}=\frac{1}{2}\mathsf{d}_{SF}^{2}(p,q)$ and,
since $q\in\partial\Omega$, it also satisfies the inequality in (72). Thus, we
may apply Lemma 4.6 and conclude the proof. ∎

Collecting all the previous results, we are in position to prove the following
theorem concerning the regularity of the normal exponential map.

######  Theorem 4.8.

Let $M$ be a smooth sub-Finsler manifold. The restriction of the sub-Finsler
normal exponential map to $D_{\epsilon}$, namely
$E|_{D_{\epsilon}}:D_{\epsilon}\to\\{0<\delta<\epsilon\\}\cap\Omega$, defines
a diffeomorphism on an open and dense subset $\mathcal{O}\subset
D_{\epsilon}$. Moreover, $\delta$ is smooth on
$E(\mathcal{O})\subset\\{0<\delta<\epsilon\\}\cap\Omega$, which is open and
with full-measure.

###### Proof.

We are going to show that $d_{(q,\lambda)}E$ is invertible for every
$(q,\lambda)$ in a suitable subset of $D_{\epsilon}$. By Corollary 4.7,
letting $U\subset\\{0<\delta<\epsilon\\}\cap\Omega$ be the set where $\delta$
is twice-differentiable, the map

$\Phi:U\to\operatorname{Ann}(\partial\Omega);\qquad\Phi(p)=e^{-\vec{H}}(d_{p}\delta)$
(74)

is a right-inverse for the normal exponential map, namely
$E\circ\Phi=\operatorname{id}_{U}$. Note that
$\Phi(U)\subset\operatorname{Ann}(\partial\Omega)$ by Corollary 4.7, in
combination with the transversality condition (63). Moreover, recalling that
the Hamiltonian is constant along the motion, we also have:

$\sqrt{2H(\Phi(p))}=\ell(\gamma)=\delta(p)\in(0,\epsilon),$ (75)

so that $\Phi(U)\subset D_{\epsilon}$. But now by the choice of the set $U$,
$\delta$ is twice-differentiable on this set and it has a Taylor expansion up
to order $2$. Thus, expanding the identity $E\circ\Phi=\operatorname{id}_{U}$
at a point $p=E(q,\lambda)$, we deduce that $d_{(q,\lambda)}E$ must be
invertible for every $(q,\lambda)\in\Phi(U)\subset D_{\epsilon}$, and thus $E$
is a local diffeomorphism around every point in $\Phi(U)$. Furthermore,
observing that $\Phi(U)$ is dense in $D_{\epsilon}$, we see that $E$ is a
local diffeomorphism everywhere on a open and dense subset $\mathcal{O}\subset
D_{\epsilon}$, containing $\Phi(U)$. Hence, we conclude that
$E|_{\mathcal{O}}$ is a diffeormorphism onto its image, being a local
diffeomorphism that is also invertible, thanks to Lemma 4.4. Finally, in order
to prove that $\delta$ is smooth on
$E(\mathcal{O})\subset\\{0<\delta<\epsilon\\}\cap\Omega$, it is enough to
observe that, by construction,

$\delta(E(q,\lambda))=\sqrt{2H(\lambda)},\qquad\forall\,(q,\lambda)\in
D_{\epsilon}.$ (76)

On $D_{\epsilon}$, $H$ is smooth, hence we conclude that $\delta$ is smooth on
$E(\mathcal{O})$. Now $U\subset E(\mathcal{O})$ so that $E(\mathcal{O})$ is
open, dense and has full-measure in $\\{0<\delta<\epsilon\\}\cap\Omega$. ∎

An immediate consequence of the previous theorem is the existence of many
geodesics that are strongly normal in the sense of Definition 4.1.

######  Corollary 4.9.

Let $M$ be a smooth sub-Finsler manifold and let $\Omega\subset M$ be an open
and bounded subset. Assume that $\partial\Omega$ is smooth and without
characteristic points. Let $p\in
E(\mathcal{O})\subset\\{0<\delta<\epsilon\\}\cap\Omega$ and let
$\gamma:[0,1]\to M$ be the unique geodesic joining $p$ and $\partial\Omega$
and realizing $\delta$. Then, $\gamma$ is strongly normal.

###### Proof.

Let $q:=\gamma(0)\in\partial\Omega$ the endpoint of $\gamma$ on the boundary
of $\Omega$. Then, since $q$ is non-characteristic point, Lemma 4.3 ensures
that $\gamma|_{[0,s]}$ can not have an abnormal lift. Hence, $\gamma$ is left
strongly normal. In order to prove that $\gamma$ is also right strongly
normal, we reason in a similar way but with $\\{\delta=\delta(p)\\}$ in place
of $\partial\Omega$. Indeed, since $\delta$ is smooth on the open set
$E(\mathcal{O})$ by Theorem 4.8 and $d_{\bar{p}}\delta$ is not vanishing for
every $\bar{p}\in E(\mathcal{O})$ as a consequence of Corollary 4.7, the set
$\Sigma:=\\{\delta=\delta(p)\\}$ defines a smooth hypersurface in a
neighborhood of the point $p$. In addition,
$\delta_{\Sigma}(q):=\mathsf{d}_{SF}(q,\Sigma)=\delta(p)$ and $\gamma$ is the
unique geodesic realizing $\delta_{\Sigma}$. Finally, applying once again
Lemma 4.3, we also deduce that $p\notin C(\Sigma)$, so repeating the argument
we did before, we conclude that $\gamma$ must be also right strongly normal.
This concludes the proof. ∎

######  Remark 4.10.

Since $E(\mathcal{O})$ has full measure in
$\\{0<\delta<\epsilon\\}\cap\Omega$, we can find $(q,\lambda)\in\mathcal{O}$
such that, denoting by $\gamma:[0,1]\to M$ the corresponding geodesic
minimizing $\delta$, we have that $\gamma(t)\in E(\mathcal{O})$ for
$\mathscr{L}^{1}$-a.e. $t\in[0,1]$. This means that $\mathscr{L}^{1}$-almost
every level set defines locally a hypersurface and, recalling that
restrictions of abnormal geodesics are still abnormal, the proof of Corollary
4.9 can be repeated to show that the curve $\gamma$ does not contain abnormal
sub-segments.

######  Theorem 4.11 (Existence of strongly normal geodesics without abnormal
sub-segments).

Let $M$ be a smooth sub-Finsler manifold. Then, there exists a strongly normal
geodesic $\gamma:[0,1]\to M$, which does not contain abnormal sub-segments.

###### Proof.

Note that Theorem 4.8 was stated for a hypersurface that is the boundary of
non-characteristic domain $\Omega$. However, without substantial
modifications, one can prove that an analogous result holds locally around a
non-characteristic point of a given smooth hypersurface $\Sigma\subset M$. In
particular, letting $q\in\Sigma\setminus C(\Sigma)$, there exists
$V_{q}\subset\Sigma$ open neighborhood of $q$, and $\epsilon>0$ such that,
denoting by

$\tilde{D}_{\epsilon}:=\\{(\bar{q},\lambda):\bar{q}\in
V_{q},\,0<\sqrt{2H(\lambda)}<\epsilon\\},$ (77)

the map $E|_{\tilde{D}_{\epsilon}}:\tilde{D}_{\epsilon}\to
E(\tilde{D}_{\epsilon})\subset\\{0<\delta_{\Sigma}<\epsilon\\}$ is a
diffeomorphism on an open and dense subset
$\mathcal{O}\subset\tilde{D}_{\epsilon}$ and $\delta_{\Sigma}$ is smooth on
$E(\mathcal{O})$. Now, Corollary 4.9 shows that there exists a point $p\in
E(\mathcal{O})$ such that the unique geodesic $\gamma:[0,1]\to M$ minimizing
$\delta$ is strongly normal and, also according to Remark 4.10, it does not
contain abnormal sub-segments. In order to conclude, we need to show that
there exists a hypersurface $\Sigma$ with $\Sigma\setminus
C(\Sigma)\neq\emptyset$. But this is a consequence of the Hörmander condition,
indeed if $\mathcal{D}_{q}\subset\Sigma_{q}$ for every $q\in\Sigma$, then
Frobenius’ theorem would ensure $\mathcal{D}$ be involutive and thus it would
not be bracket-generating. ∎

### 4.2 Regularity of the distance function

We state below the definition of conjugate and cut loci in a sub-Finsler
manifold, following the blueprint of the sub-Riemannian setting, cf. [ABB20,
Chap. 11] or [BR19].

######  Definition 4.12 (Conjugate point).

Let $M$ be a smooth sub-Finsler manifold and let $\gamma:[0,1]\to M$ be a
normal geodesic with initial covector $\lambda\in T^{*}_{p}M$, that is
$\gamma(t)=\exp_{p}(t\lambda)$. We say that $q=\exp_{p}(\bar{t}\lambda)$ is a
_conjugate point_ to $p$ along $\gamma$ if $\bar{t}\lambda$ is a critical
point for $\exp_{p}$.

######  Definition 4.13 (Cut locus).

Let $M$ be a smooth sub-Finsler manifold and let $p\in M$. We say that $q\in
M$ is a _smooth point_ with respect to $p$, and write $q\in\Sigma_{p}$, if
there exists a unique geodesic $\gamma:[0,1]\to M$ joining $p$ and $q$, which
is not abnormal and such that $q$ is not conjugate along $p$. Define the _cut
locus_ of $p\in M$ as ${\rm Cut}(p):=M\setminus\Sigma_{p}$. Finally, the cut
locus of $M$ is the set

${\rm Cut}(M):=\\{(p,q)\in M\times M:q\in{\rm Cut}(p)\\}\subset M\times M.$
(78)

######  Remark 4.14.

In the sub-Riemannian setting, according to [Agr09], the set of smooth points
with respect to $p$ is open and dense. However, it is an open question to
understand whether its complement, that is the cut locus, is negligible.

Outside the cut locus of a sub-Finsler manifold, we can define the _$t$
-midpoint map_, for $t\in[0,1]$, as the map $\phi_{t}:M\times M\setminus{\rm
Cut}(M)\to M$ assigning to $(p,q)$ the $t$-midpoint of the (unique) geodesic
$\gamma_{p,q}$ joining $p$ and $q$. More precisely, for every $(p,q)\in
M\times M\setminus{\rm Cut}(M)$,

$\phi_{t}(p,q):=e_{t}(\gamma_{p,q})=\exp_{q}((t-1)\lambda_{p}),\qquad\text{where
}\lambda_{p}\in T^{*}_{q}M\text{ such that }p=\exp_{q}(-\lambda_{p}).$ (79)

Note that, by definition of cut locus, the $t$-midpoint map is well-defined
since the geodesic joining $p$ and $q$ for $q\notin{\rm Cut}(p)$ is unique and
strictly normal, i.e. without abnormal lifts.

We report a useful result relating the regularity of the squared distance
function on a sub-Finsler manifold $M$ with the cut locus. Such result can be
proved repeating verbatim the proof of [ABB20, Prop. 11.4], in light of
Proposition 3.14 and Lemma 4.6. For every $p\in M$, let
$\mathfrak{f}_{p}:=\frac{1}{2}\mathsf{d}_{SF}^{2}(\cdot,p)$.

######  Proposition 4.15.

Let $M$ be a smooth sub-Finsler manifold and let $p,q\in M$. Assume there
exists an open neighborhood $\mathcal{O}_{q}\subset M$ of $q$ such that
$\mathfrak{f}_{p}$ is smooth. Then, $\mathcal{O}_{q}\subset\Sigma_{p}$ and

$\phi_{t}(p,z)=\exp_{z}((t-1)d_{z}\mathfrak{f}_{p}),\qquad\forall\,z\in\mathcal{O}_{q}.$
(80)

Thanks to Proposition 4.15, the regularity of the squared distance ensures
uniqueness of geodesics and smoothness of the $t$-midpoint map. Thus, it is
desirable to understand where the squared distance is smooth. In this regard,
[ABR18, Thm. 2.19] proves the regularity of the squared distance function
along left strongly normal geodesics. We refer to [ABR18, App. A] for further
details.

######  Theorem 4.16 ([ABR18, Thm. 2.19]).

Let $M$ be a smooth sub-Finsler manifold and let $\gamma:[0,1]\to M$ be a left
strongly normal geodesic. Then there exists $\epsilon>0$ and an open
neighborhood $U\subset M\times M$ such that:

  1. (i)

$(\gamma(0),\gamma(t))\in U$ for all $t\in(0,\epsilon)$;

  2. (ii)

For any $(p,q)\in U$ there exists a unique (normal) geodesic joining $p$ and
$q$, shorter than $\epsilon$;

  3. (iii)

The squared distance function $(p,q)\mapsto\mathsf{d}_{SF}^{2}(p,q)$ is smooth
on $U$.

The regularity of the squared distance function can be “propagated” along
geodesics that do not admit abnormal sub-segments, applying the previous
theorem for every sub-segment.

######  Corollary 4.17.

Let $M$ be a smooth sub-Finsler manifold and let $\gamma:[0,1]\to M$ be a
geodesic that does not admit abnormal sub-segments. Then, for every
$s\in[0,1]$, there exists $\epsilon>0$ and an open neighborhood $U\subset
M\times M$ such that:

  1. (i)

$(\gamma(s),\gamma(t))\in U$ for all $t\in[0,1]$ such that $0<|t-s|<\epsilon$;

  2. (ii)

For any $(p,q)\in U$ there exists a unique (normal) geodesic joining $p$ and
$q$, shorter than $\epsilon$;

  3. (iii)

The squared distance function $(p,q)\mapsto\mathsf{d}_{SF}^{2}(p,q)$ is smooth
on $U$.

### 4.3 Volume contraction rate along geodesics

Our goal is to quantify the contraction rate of small volumes along geodesics.
To do this, we combine the smoothness of the $t$-midpoint map, with a lower
bound on the so-called geodesic dimension. The latter has been introduced in
[ABR18] for sub-Riemannian manifolds and in [Riz16, Def. 5.47] for general
metric measure spaces. We recall below the definition.

Let $M$ be a smooth sub-Finsler manifold. Given a point $p\in M$ and a Borel
set $\Omega\subset M\setminus{\rm Cut}(p)$, we define the _geodesic homothety_
of $\Omega$ with center $p$ and ratio $t\in[0,1]$ as

$\Omega_{t}^{p}:=\\{\phi_{t}(p,q):q\in\Omega\\}.$ (81)

In the sequel, we say that $\mathfrak{m}$ is a _smooth measure_ if, in
coordinates, is absolutely continuous with respect the Lebesgue measure of the
chart with a smooth and positive density. We will consider the metric measure
space $(M,\mathsf{d}_{SF},\mathfrak{m})$.

######  Definition 4.18.

Let $M$ be a smooth sub-Finsler manifold, equipped with a smooth measure
$\mathfrak{m}$. For any $p\in M$ and $s>0$, define

$C_{s}(p):=\sup\left\\{\limsup_{t\to
0}\frac{1}{t^{s}}\frac{\mathfrak{m}(\Omega_{t}^{p})}{\mathfrak{m}(\Omega)}:\Omega\subset
M\setminus{\rm Cut}(p)\text{ Borel, bounded and
}\mathfrak{m}(\Omega)\in(0,+\infty)\right\\},$ (82)

We define the _geodesic dimension_ of $(M,\mathsf{d}_{SF},\mathfrak{m})$ at
$p\in M$ as the non-negative real number

$\mathcal{N}(p):=\inf\\{s>0:C_{s}(p)=+\infty\\}=\sup\\{s>0:C_{s}(p)=0\\},$
(83)

with the conventions $\inf\emptyset=+\infty$ and $\sup\emptyset=0$.

######  Remark 4.19.

In [Riz16], the definition of geodesic dimension is given for metric measure
spaces with _negligible cut loci_. While, in [BMR22, Def. 4.17], this
assumption is dropped and the definition of geodesic dimension is given by
taking the supremum (82) over Borel sets. In our work, we modified the
definition by considering the supremum (82) over sets $\Omega$ outside the cut
locus ${\rm Cut}(p)$. This may not be equivalent to neither of the previous
definitions but it is tailored to our specific purposes.

We now prove a fundamental theorem which relates the geodesic and topological
dimensions of a sub-Finsler manifold $M$. This result is a suitable adaptation
of [Riz16, Thm. 4] to our setting.

######  Proposition 4.20.

Let $M$ be a smooth sub-Finsler manifold, equipped with a smooth measure
$\mathfrak{m}$. Assume that $r(p)<n:=\dim M$ for every $p\in M$. Then,

$\mathcal{N}(p)\geq n+1,\qquad\forall\,p\in M.$ (84)

###### Proof.

Let $\mathsf{d}_{SR}$ be a sub-Riemannian distance on the manifold $M$,
equivalent to $\mathsf{d}_{SF}$ (see (31)). The Ball-Box theorem, cf. [Jea14,
Cor. 2.1], ensures that for every $p\in M$ there exist $n_{p}\geq n+1$ and a
positive constant $C_{p}$ such that

$\mathfrak{m}\big{(}B_{r}^{SR}(p)\big{)}\leq C_{p}\cdot r^{n_{p}}\qquad\text{
for $r$ sufficiently small.}$ (85)

Since $\mathsf{d}_{SF}$ and $\mathsf{d}_{SR}$ are equivalent, up to changing
the constant, the same estimate holds for sub-Finsler balls, in particular

$\limsup_{r\to 0}\frac{\mathfrak{m}\big{(}B_{r}^{SF}(p)\big{)}}{r^{k}}=0$ (86)

for every $k<n+1$. Take any $\Omega\subset M\setminus{\rm Cut}(p)$ Borel,
bounded and with $\mathfrak{m}(\Omega)\in(0,+\infty)$ and consider $R>0$ such
that $\Omega\subset B_{R}^{SF}(p)$. Note that $\Omega_{t}^{p}\subset
B_{tR}^{SF}(p)$ and thus for every $k<n+1$ we have that

$\limsup_{t\to
0}\frac{\mathfrak{m}(\Omega_{t}^{p})}{t^{k}\mathfrak{m}(\Omega)}\leq\limsup_{t\to
0}\frac{\mathfrak{m}\big{(}B_{tR}^{SF}(p)\big{)}}{t^{k}\mathfrak{m}(\Omega)}=\limsup_{t\to
0}\frac{\mathfrak{m}\big{(}B_{tR}^{SF}(p)\big{)}}{(tR)^{k}}\cdot\frac{R^{k}}{\mathfrak{m}(\Omega)}=0,$
(87)

where we used (86) for the last equality. Since $\Omega$ was arbitrary, we
deduce that $C_{k}(p)=0$ for every $k<n+1$ and then $\mathcal{N}(p)\geq n+1$.
∎

######  Remark 4.21.

For an equiregular sub-Finsler manifold, with the same proof, it is possible
to improve the estimate of Proposition 4.20. In fact, in this case the Ball-
Box theorem provides the estimate (85) with $n_{p}$ equal to the Hausdorff
dimension $\dim_{H}(M)$, for every $p$, and consequently
$\mathcal{N}(p)\geq\dim_{H}(M)$, cf. [ABR18, Prop. 5.49].

By construction, the geodesic dimension controls the contraction rate of
volumes along geodesics. This information can be transferred to the
$t$-midpoint map, provided that is smooth. By invoking Theorem 4.16, we can
always guarantee the smoothness of the $t$-midpoint map for a sufficiently
short segment of a geodesic without abnormal sub-segments.

######  Theorem 4.22.

Let $M$ be a smooth sub-Finsler manifold equipped with a smooth measure
$\mathfrak{m}$ and such that $r(p)<n:=\dim M$ for every $p\in M$. Let
$\gamma:[0,1]\to M$ be a geodesic that does not admit abnormal sub-segments,
with endpoints $p$ and $q$. Assume that $(p,q)$ belongs to the open set $U$,
found in Theorem 4.16. Then, either
$|\det\big{(}d_{q}\phi_{t}(p,\cdot)\big{)}|$ has infinite order at $t=0$ or

$|\det\big{(}d_{q}\phi_{t}(p,\cdot)\big{)}|\sim t^{m_{p}},\qquad\text{as }t\to
0$ (88)

for some integer $m_{p}\geq\mathcal{N}(p)\geq n+1$.

###### Proof.

Since, by assumption $(p,q)\in U$, we can apply item (iii) of Theorem 4.16,
deducing the regularity of the distance function. Combining this with
Proposition 4.15 and the homogeneity of the Hamiltonian flow, there exists an
open neighborhood $V\subset M$ of $q$, such that the function

$[0,1)\times V\ni(t,z)\mapsto
d_{z}\phi_{t}(p,\cdot)=d_{z}\left(\exp_{z}((t-1)d_{z}\mathfrak{f}_{p})\right)$
(89)

is smooth. Thus, we can compute the Taylor expansion of its determinant in the
$t$-variable at order $N:=\lceil\mathcal{N}(p)\rceil-1<\mathcal{N}(p)$,
obtaining:

$\det\big{(}d_{z}\phi_{t}(p,\cdot)\big{)}=\sum_{i=0}^{N}a_{i}(z)t^{i}+t^{N+1}R_{N}(t,z),\qquad\forall\,z\in
V,$ (90)

where the functions $a_{i}$ and $R_{N}$ are smooth. Arguing by contradiction,
we assume that there exists $j\leq N$ such that $a_{j}(q)\neq 0$ and define

$m:=\min\\{i\leq N\,:\,\exists\,z\in V\text{ such that }a_{i}(z)\neq 0\\}.$
(91)

Note that $m\leq j$ since $a_{j}(q)\neq 0$ and thus $m\leq N$. Without loss of
generality, we can assume that $V$ and $p$ are contained in the same
coordinate chart and that $a_{m}>0$ on an open subset $\tilde{V}\subset V$
with positive measure. Then, in charts, it holds that

$\mathscr{L}^{n}\big{(}\tilde{V}_{t}^{p}\big{)}=\int_{\tilde{V}}\big{|}\det\big{(}d_{z}\phi_{t}(p,\cdot)\big{)}\big{|}\,\mathrm{d}z=\int_{\tilde{V}}a_{m}(z)\,\mathrm{d}z\cdot
t^{m}+o(t^{m})\qquad\text{as }t\to 0.$ (92)

Therefore, recalling that $\mathfrak{m}$ is a smooth measure, there exists a
constant $a>0$ such that

$\mathfrak{m}\big{(}\tilde{V}_{t}^{p}\big{)}\geq a\cdot t^{m},$ (93)

for every $t$ sufficiently small. As a consequence, taking any
$s\in(N,\mathcal{N}(p))$ we have that

$\limsup_{t\to
0}\frac{1}{t^{s}}\frac{\mathfrak{m}(\tilde{V}_{t}^{p})}{\mathfrak{m}(\tilde{V})}\geq\limsup_{t\to
0}\frac{1}{\mathfrak{m}(\tilde{V})}\frac{a\cdot t^{m}}{t^{s}}=+\infty,$ (94)

and therefore we deduce $C_{s}(p)=+\infty$, which in turn implies
$\mathcal{N}(p)\leq s$, giving a contradiction. ∎

Theorem 4.22 motivates the following definition.

######  Definition 4.23 (Ample geodesic).

Let $M$ be a smooth sub-Finsler manifold and let $\gamma:[0,1]\to M$ be a
strictly normal geodesic not admitting abnormal sub-segments. We say that
$\gamma$ is _ample_ if, for every couple of distinct points
$p,q\in\gamma([0,1])$, $|\det\big{(}d_{q}\phi_{t}(p,\cdot)\big{)}|$ exists and
has finite order in $t=0$.

######  Remark 4.24.

The concept of ample geodesic in the sub-Riemannian setting has been
introduced in [ABR18] and it differs from Definition 4.23. However, we remark
that, in sub-Riemannian manifolds, for ample geodesics in the sense of
[ABR18], $|\det\big{(}d_{q}\phi_{t}(p,\cdot)\big{)}|$ has finite order equal
to the geodesic dimension at $p$, cf. [ABR18, Lem. 6.27]. Thus, our definition
is weaker, but enough for our purposes.

### 4.4 Proof of Theorem 1.5

Let $M$ be a smooth sub-Finsler manifold and let $\phi_{t}$ the $t$-midpoint
map, defined as in (79). For ease of notation, set

$\mathcal{M}(p,q):=\phi_{1/2}(p,q),\qquad\forall\,(p,q)\in M\times
M\setminus{\rm Cut}(M),$ (95)

be the $1/2$-midpoint map or simply midpoint map. Reasoning as in [Jui21,
Prop. 3.1], we obtain the following result as a consequence of Corollary 4.17
and Theorem 4.22. This argument hinges upon Theorem 4.11, which establishes
the existence of a geodesic without abnormal sub-segments in a sub-Finsler
manifold.

######  Proposition 4.25.

Let $M$ be a smooth sub-Finsler manifold equipped with a smooth measure
$\mathfrak{m}$ and such that $r(p)<n:=\dim M$ for every $p\in M$. Let
$\gamma:[0,1]\to M$ be the geodesic identified in Theorem 4.11 and let
$\varepsilon>0$. Then, there exist $0\leq a<b\leq 1$ such that, letting
$\bar{p}:=\gamma(a)$, $\bar{q}:=\gamma(b)$, the following statements hold:

  1. (i)

$\bar{p}\notin{\rm Cut}(\bar{q})$, $\bar{q}\notin{\rm Cut}(\bar{p})$ and, for
every $t\in(a,b)$, we have $\bar{p},\bar{q}\notin{\rm Cut}(\gamma(t))$.
Moreover, for every $t\in(a,b)$, $\mathfrak{f}_{\gamma(t)}$ is smooth in a
neighborhood of $\bar{p}$ and in a neighborhood $\bar{q}$.

  2. (ii)

If, in addition, $\gamma$ is ample, the midpoint map satisfies

$|\det
d_{\bar{q}}\mathcal{M}(\bar{p},\cdot)|\leq(1+\varepsilon)2^{-m_{\bar{p}}},\qquad|\det
d_{\bar{p}}\mathcal{M}(\cdot,\bar{q})|\leq(1+\varepsilon)2^{-m_{\bar{q}}}$
(96)

where $m_{\bar{p}}$ and $m_{\bar{q}}$ are defined by (88) and
$m_{\bar{p}},m_{\bar{q}}\geq n+1$.

Given $z\in M$, define the _inverse geodesic map_
$\mathcal{I}_{z}:M\setminus{\rm Cut}(z)\to M$ as

$\mathcal{I}_{z}(p)=\exp_{z}(-\lambda)\qquad\text{where }\lambda\in
T^{*}_{z}M\text{ such that }p=\exp_{z}(\lambda).$ (97)

We may interpret this map as the one associating to $p$ the point
$\mathcal{I}_{z}(p)$ such that $z$ is the midpoint of $x$ and
$\mathcal{I}_{z}(p)$.

We prove now the main theorem of this section, which also implies Theorem 1.5.
Our strategy is an adaptation to the sub-Finsler setting of the one proposed
in [Jui21].

######  Theorem 4.26.

Let $M$ be a complete smooth sub-Finsler manifold equipped with a smooth
measure $\mathfrak{m}$ and such that $r(p)<n:=\dim M$ for every $p\in M$.
Then, the metric measure space $(M,\mathsf{d}_{SF},\mathfrak{m})$ does not
satisfy the Brunn–Minkowski inequality $\mathsf{BM}(K,N)$, for every
$K\in\mathbb{R}$ and $N\in(1,\infty)$.

###### Proof.

Fix $\varepsilon>0$, $K\in\mathbb{R}$ and $N\in(1,\infty)$. Let
$\gamma:[0,1]\to M$ be the geodesic identified by Theorem 4.11 and assume it
is contained in a coordinate chart with (sub-Finsler) diameter $D>0$. Up to
restricting the domain of the chart and the geodesic, we can also assume that

$(1-\varepsilon)\mathscr{L}^{n}\leq\mathfrak{m}\leq(1+\varepsilon)\mathscr{L}^{n}\qquad\text{and}\qquad\tau_{K,N}^{(1/2)}(\theta)\geq\frac{1}{2}-\varepsilon,\quad\forall\theta\leq
D,$ (98)

where the second inequality can be fulfilled, according to Remark 2.1.
Moreover, let $0\leq a<b\leq 1$ be as in Proposition 4.25. We proceed by
contradiction and assume that $(M,\mathsf{d}_{SF},\mathfrak{m})$ satisfies the
$\mathsf{BM}(K,N)$.

First of all, suppose that $\gamma$ is not ample. According to [MPR22b, Prop.
5.3], the Brunn–Minkowski inequality $\mathsf{BM}(K,N)$ implies the
$\mathsf{MCP}(K,N)$ condition333In that paper, the proposition is proved for
essentially non-branching metric measure spaces. However, what is needed is a
measurable selection of geodesics which we have locally around the curve
$\gamma$.. Therefore, $(M,\mathsf{d}_{SF},\mathfrak{m})$ satisfies the
$\mathsf{MCP}(K,N)$ condition and, for the moment, assume $K=0$. Set
$\bar{p}:=\gamma(a)$ and $\bar{q}:=\gamma(b)$ and let
$\Omega_{\varrho}:=B_{\varrho}(\bar{q})$ for $\varrho>0$. From the
$\mathsf{MCP}(0,N)$ condition we get

$\mathfrak{m}\big{(}\Omega_{\varrho,t}^{\bar{p}}\big{)}\geq
t^{N}\mathfrak{m}(\Omega_{\varrho}),\qquad\forall\,t\in[0,1],\,\varrho>0.$
(99)

If $\varrho$ is sufficiently small, then
$\Omega_{\varrho,t}^{\bar{p}}=\phi_{t}(\bar{p},\Omega_{\varrho})$ for
$t\in[0,1)$, therefore, employing the first estimate in (98), the inequality
(99) can be reformulated as follows:

$\frac{1+\varepsilon}{1-\varepsilon}\fint_{\Omega_{\varrho}}|\det\big{(}d_{z}\phi_{t}(\bar{p},\cdot)\big{)}|\,\mathrm{d}z\geq\frac{\mathfrak{m}(\Omega_{\varrho,t}^{\bar{p}})}{\mathfrak{m}(\Omega_{\varrho})}\geq
t^{N},\qquad\forall\,t\in[0,1),\,\varrho>0.$ (100)

Taking the limit as $\varrho\to 0$, and then the limit as $t\to 0$, we find
that the order of
$\big{|}\det\big{(}d_{\bar{q}}\phi_{t}(\bar{p},\cdot)\big{)}\big{|}$ should be
smaller than or equal to $N$, giving a contradiction. Finally, if $K\neq 0$,
observe that the behavior of the distortion coefficients, as $t\to 0$, is
comparable with $t$, namely there exists a constant
$C=C(K,N,\mathsf{d}(\bar{p},\bar{q}))>0$ such that

$\tau_{K,N}^{(t)}(\theta)\geq Ct,\qquad\text{as }t\to
0,\qquad\forall\,\theta\in(\mathsf{d}(\bar{p},\bar{q})-\varrho,\mathsf{d}(\bar{p},\bar{q})+\varrho).$
(101)

Therefore, repeating the same argument that we did for the case $K=0$, we
obtain the sought contradiction.

Suppose instead that the geodesic $\gamma$ is ample and let $m$ be the unique
midpoint between $\bar{p}=\gamma(a)$ and $\bar{q}=\gamma(b)$. According to
item (i) of Proposition 4.25, the map $\mathcal{I}_{m}$ is well-defined and
smooth in a neighborhood of $\bar{p}$ and $\bar{q}$, moreover by definition
$\mathcal{I}_{m}(\bar{q})=\bar{p}$ and $\mathcal{I}_{m}(\bar{p})=\bar{q}$.
Note that $\mathcal{I}_{m}\circ\mathcal{I}_{m}=\operatorname{id}$ (where
defined), thus

$|\det(d_{\bar{p}}\mathcal{I}_{m})|\cdot|\det(d_{\bar{q}}\mathcal{I}_{m})|=\big{|}\det\big{(}d_{\bar{q}}(\mathcal{I}_{m}\circ\mathcal{I}_{m})\big{)}\big{|}=1.$
(102)

Therefore, at least one between $|\det(d_{\bar{q}}\mathcal{I}_{m})|$ and
$|\det(d_{\bar{p}}\mathcal{I}_{m})|$ is greater than or equal to 1, without
loss of generality we assume

$|\det(d_{\bar{q}}\mathcal{I}_{m})|\geq 1.$ (103)

Let $B_{\varrho}:=B^{eu}_{\varrho}(\bar{q})$ the (Euclidean) ball of radius
$\varrho>0$ centered in $\bar{q}$. Introduce the function $F:B_{\varrho}\times
B_{\varrho}\to M$, defined as

$B_{\varrho}\times B_{\varrho}\ni(x,y)\mapsto
F(x,y):=\mathcal{M}(\mathcal{I}_{m}(x),y).$

Observe that, for $\varrho$ small enough, $F$ is well-defined and by
construction $F(x,x)=m$ for every $x\in B_{\varrho}$. Therefore, we deduce
that for every vector $v\in T_{\bar{q}}M\cong\mathbb{R}^{n}$, the following
holds:

$0=d_{(\bar{q},\bar{q})}F(v,v)=\big{(}d_{\bar{p}}\mathcal{M}(\cdot,\bar{q})\circ
d_{\bar{q}}\mathcal{I}_{m}\big{)}\,v+d_{\bar{q}}\mathcal{M}(\bar{p},\cdot)\,v.$
(104)

Since the former identity is true for every vector $v\in\mathbb{R}^{n}$, we
can conclude that

$d_{\bar{p}}\mathcal{M}(\cdot,\bar{q})\circ
d_{\bar{q}}\mathcal{I}_{m}+d_{\bar{q}}\mathcal{M}(\bar{p},\cdot)=0,$ (105)

and consequently, for every $v,w\in\mathbb{R}^{n}$, we have

$d_{(\bar{q},\bar{q})}F(v,w)=\big{(}d_{\bar{p}}\mathcal{M}(\cdot,\bar{q})\circ
d_{\bar{q}}\mathcal{I}_{m}\big{)}\,v+d_{\bar{q}}\mathcal{M}(\bar{p},\cdot)\,w=d_{\bar{q}}\mathcal{M}(\bar{p},\cdot)\,(w-v).$
(106)

In particular, we obtain a Taylor expansion of the function $F$ at the point
$(\bar{q},\bar{q})$ that in coordinates takes the form:

$\left\lVert
F(\bar{q}+v,\bar{q}+w)-m-d_{\bar{q}}\mathcal{M}(\bar{p},\cdot)\,(w-v)\right\rVert_{eu}=o(\left\lVert
v\right\rVert_{eu}+\left\lVert w\right\rVert_{eu}),\qquad\text{as }v,w\to 0.$

Then, as $v$ and $w$ vary in $B^{eu}_{\varrho}(0)$, $v-w$ varies in
$B^{eu}_{2\varrho}(0)$, and we obtain that

$F(B_{\varrho},B_{\varrho})\subseteq
m+d_{\bar{q}}\mathcal{M}(\bar{p},\cdot)\,\big{(}B^{eu}_{2\varrho}(0)\big{)}+B^{eu}_{\omega(\varrho)}(0),$
(107)

where $\omega:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is such that $\omega(r)=o(r)$
when $r\to 0^{+}$. Now, consider $A_{\varrho}:=\mathcal{I}_{m}(B_{\varrho})$
and note that by definition
$M_{1/2}(A_{\varrho},B_{\varrho})=F(B_{\varrho},B_{\varrho})$, then using
(107) we conclude that, as $\varrho\to 0$,

$\begin{split}\mathscr{L}^{n}&\big{(}M_{1/2}(A_{\varrho},B_{\varrho})\big{)}=\mathscr{L}^{n}\big{(}F(B_{\varrho},B_{\varrho})\big{)}\leq\mathscr{L}^{n}\Big{(}d_{\bar{q}}\mathcal{M}(p,\cdot)\,\big{(}B^{eu}_{2\varrho}(0)\big{)}\Big{)}+o(\varrho^{n})\\\
&=\big{|}\det(d_{\bar{q}}\mathcal{M}(p,\cdot))\big{|}\cdot\omega_{n}2^{n}\varrho^{n}+o(\varrho^{n})\leq(1+\varepsilon)2^{n-m_{q}}\omega_{n}\varrho^{n}+o(\varrho^{n})\leq\frac{1}{2}(1+\varepsilon)\omega_{n}\varrho^{n}+o(\varrho^{n})\end{split}$

where $\omega_{n}=\mathscr{L}^{n}(B^{eu}_{1}(0))$ and the two last
inequalities follow from (96) and $m_{\bar{q}}\geq n+1$. On the other hand, it
holds that $\mathscr{L}^{n}(B_{\varrho})=\omega_{n}\varrho^{n}$ and, as
$\varrho\to 0$,

$\mathscr{L}^{n}(A_{\varrho})=\mathscr{L}^{n}(\mathcal{I}_{m}(B_{\varrho}))=\big{(}|\det(d_{\bar{q}}\mathcal{I}_{m})|+O(\varrho)\big{)}\,\mathscr{L}^{n}(B_{\varrho})\geq\omega_{n}\varrho^{n}+o(\varrho^{n}).$
(108)

Taking into account the first estimate of (98), we deduce the following
inequalities for the measure $\mathfrak{m}$, as $\varrho\to 0$,

$\begin{gathered}\mathfrak{m}\big{(}M_{1/2}(A_{\varrho},B_{\varrho})\big{)}\leq\frac{1}{2}(1+\varepsilon)^{2}\omega_{n}\varrho^{n}+o(\varrho^{n}),\\\
\mathfrak{m}(A_{\varrho})\geq(1-\varepsilon)\omega_{n}\varrho^{n}+o(\varrho^{n})\qquad\text{and}\qquad\mathfrak{m}(B_{\varrho})\geq(1-\varepsilon)\omega_{n}\varrho^{n}.\end{gathered}$

Finally, if $\varepsilon$ is small enough we can find $\varrho$ sufficiently
small such that

$\begin{split}\mathfrak{m}\big{(}M_{1/2}(A_{\varrho},B_{\varrho})\big{)}^{\frac{1}{N}}&<\bigg{(}\frac{1}{2}-\varepsilon\bigg{)}\,\mathfrak{m}(A_{\varrho})^{\frac{1}{N}}+\bigg{(}\frac{1}{2}-\varepsilon\bigg{)}\,\mathfrak{m}(B_{\varrho})^{\frac{1}{N}}\\\
&\leq\tau_{K,N}^{(1/2)}\big{(}\Theta(A_{\varrho},B_{\varrho})\big{)}\,\mathfrak{m}(A_{\varrho})^{\frac{1}{N}}+\tau_{K,N}^{(1/2)}\big{(}\Theta(A_{\varrho},B_{\varrho})\big{)}\,\mathfrak{m}(B_{\varrho})^{\frac{1}{N}},\end{split}$
(109)

which contradicts the Brunn–Minkowski inequality $\mathsf{BM}(K,N)$. ∎

######  Remark 4.27.

Observe that the argument presented in this section is local, around the
geodesic without abnormal sub-segments. Thus, repeating the same proof, we can
extend Theorem 4.26 if the assumption on the rank holds on an open set
$V\subset M$, namely $r(p)<n$ for every $p\in V$.

## 5 Failure of the $\mathsf{CD}(K,N)$ condition in the sub-Finsler Heisenberg
group

In this section, we disprove the curvature-dimension condition in the sub-
Finsler Heisenberg group, cf. Theorem 1.8. Our strategy relies on the explicit
expression of geodesics in terms of convex trigonometric functions, found in
[Lok21].

### 5.1 Convex trigonometry

In this section, we recall the definition and main properties of the convex
trigonometric functions, firstly introduced in [Lok19]. Let
$\Omega\subset\mathbb{R}^{2}$ be a convex, compact set, such that
$O:=(0,0)\in\text{Int}(\Omega)$ and denote by $\mathbb{S}$ its surface area.

######  Definition 5.1.

Let $\theta\in\mathbb{R}$ denote a generalized angle. If
$0\leq\theta<2\mathbb{S}$ define $P_{\theta}$ as the point on the boundary of
$\Omega$, such that the area of the sector of $\Omega$ between the rays $Ox$
and $OP_{\theta}$ is $\frac{1}{2}\theta$ (see Figure 2). Moreover, define
$\sin_{\Omega}(\theta)$ and $\cos_{\Omega}(\theta)$ as the coordinates of the
point $P_{\theta}$, i.e.

$P_{\theta}=\big{(}\sin_{\Omega}(\theta),\cos_{\Omega}(\theta)\big{)}.$

Finally, extend these trigonometric functions outside the interval
$[0,2\mathbb{S})$ by periodicity (of period $2\mathbb{S}$), so that for every
$k\in\mathbb{Z}$.

$\cos_{\Omega}(\theta)=\cos_{\Omega}(\theta+2k\mathbb{S}),\quad\sin_{\Omega}(\theta)=\sin_{\Omega}(\theta+2k\mathbb{S})\quad\text{and}\quad
P_{\theta}=P_{\theta+2k\mathbb{S}}.$

Observe that by definition $\sin_{\Omega}(0)=0$ and that when $\Omega$ is the
Euclidean closed unit ball we recover the classical trigonometric functions.

${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sin_{\Omega}(\theta)}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\cos_{\Omega}(\theta)}}$$O$$\frac{1}{2}\theta$$\Omega$$P_{\theta}$
Figure 1: Values of the generalized trigonometric functions $\cos_{\Omega}$
and $\sin_{\Omega}$.

$O$$\frac{1}{2}\theta$$\Omega$$P_{\theta}$$Q_{\psi}$ Figure 2: Representation
of the correspondence $\theta\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega}\psi$.

Consider now the polar set:

$\Omega^{\circ}:=\\{(p,q)\in\mathbb{R}^{2}\,:\,px+qy\leq 1\text{ for every
}(x,y)\in\Omega\\},$

which is itself a convex, compact set such that
$O\in\text{Int}(\Omega^{\circ})$. Therefore, we can consider the trigonometric
functions $\sin_{\Omega^{\circ}}$ and $\cos_{\Omega^{\circ}}$. Observe that,
by definition of polar set, it holds that

$\cos_{\Omega}(\theta)\cos_{\Omega^{\circ}}(\psi)+\sin_{\Omega}(\theta)\sin_{\Omega^{\circ}}(\psi)\leq
1,\qquad\text{for every }\theta,\psi\in\mathbb{R}.$ (110)

######  Definition 5.2.

We say that two angles $\theta,\psi\in\mathbb{R}$ _correspond_ to each other
and write $\theta\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega}\psi$ if the vector
$Q_{\psi}:=(\cos_{\Omega^{\circ}}(\psi),\sin_{\Omega^{\circ}}(\psi))$
determines a half-plane containing $\Omega$ (see Figure 2).

By the bipolar theorem [Roc70, Thm. 14.5], it holds that
$\Omega^{\circ\circ}=\Omega$ and this allow to prove the following symmetry
property for the correspondence just defined.

######  Proposition 5.3.

Let $\Omega\subset\mathbb{R}^{2}$ be a convex and compact set, with
$O\in\text{Int}(\Omega)$. Given two angles $\theta,\psi\in\mathbb{R}$,
$\theta\ext@arrow 9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega}\psi$
if and only if $\psi\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega^{\circ}}\theta$.
Moreover, the following analogous of the Pythagorean equality holds:

$\theta\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega}\psi\qquad\text{ if and
only if
}\qquad\cos_{\Omega}(\theta)\cos_{\Omega^{\circ}}(\psi)+\sin_{\Omega}(\theta)\sin_{\Omega^{\circ}}(\psi)=1.$
(111)

The correspondence $\theta\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega}\psi$ is not one-to-one
in general, in fact if the boundary of $\Omega$ has a corner at the point
$P_{\theta}$, the angle $\theta$ corresponds to an interval of angles (in
every period). Nonetheless, we can define a monotone multi-valued map
$C^{\circ}$ that maps an angle $\theta$ to the maximal closed interval
containing angles corresponding to $\theta$. This function has the following
periodicity property:

$C^{\circ}(\theta+2\mathbb{S}k)=C^{\circ}(\theta)+2\mathbb{S}^{\circ}k\qquad\text{
for every }k\in\mathbb{Z},$

where $\mathbb{S}^{\circ}$ denotes the surface area of $\Omega^{\circ}$. If
$\Omega$ is strictly convex, then the map $C^{\circ}$ is strictly monotone,
while if the boundary of $\Omega$ is $C^{1}$, then $C^{\circ}$ is a (single-
valued) map from $\mathbb{R}$ to $\mathbb{R}$ and it is continuous.
Analogously, we can define the map $C_{\circ}$ associated to the
correspondence $\psi\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega^{\circ}}\theta$.
Proposition 5.3 guarantees that $C_{\circ}\circ C^{\circ}=C^{\circ}\circ
C_{\circ}=\text{id}$.

######  Proposition 5.4.

Let $\Omega\subset\mathbb{R}^{2}$ as above. The trigonometric functions
$\sin_{\Omega}$ and $\cos_{\Omega}$ are Lipschitz and therefore differentiable
almost everywhere. At every differentiability point $\theta$ of both
functions, there exists a unique angle $\psi$ corresponding to $\theta$ and it
holds that

$\sin_{\Omega}^{\prime}(\theta)=\cos_{\Omega^{\circ}}(\psi)\qquad\text{and}\qquad\cos_{\Omega}^{\prime}(\theta)=-\sin_{\Omega^{\circ}}(\psi).$

Naturally, the analogous result holds for the trigonometric functions
$\sin_{\Omega^{\circ}}$ and $\cos_{\Omega^{\circ}}$.

As a corollary of the previous proposition, we obtain the following convexity
properties for the trigonometric functions.

######  Corollary 5.5.

The functions $\sin_{\Omega}$ and $\cos_{\Omega}$ are concave in every
interval in which they are non-negative and convex in every interval in which
they are non-positive.

This convexity properties of the trigonometric functions will play a small but
fundamental role in Section 5.3 in the form of the following corollaries.

######  Corollary 5.6.

Given a non-null constant $k\in\mathbb{R}$ and every angle $\theta$, consider
the function

$g:\mathbb{R}\to\mathbb{R};\qquad
g(t):=\sin_{\Omega}(\theta)\cos_{\Omega}(\theta+kt)-\cos_{\Omega}(\theta)\sin_{\Omega}(\theta+kt).$
(112)

If $k>0$ this function is convex for positive values of $t$ and concave for
negative values of $t$, locally around $0$. Vice versa, If $k<0$ it is concave
for positive values of $t$ and convex for negative values of $t$, locally
around $0$.

###### Proof.

The function $g(t)$ can be seen as a scalar product of two vectors in
$\mathbb{R}^{2}$, therefore it is invariant by rotations. In particular, we
consider the rotation that sends $\theta$ to $0$: this maps $P_{\theta}$ to
the the positive $x$-axis and the set $\Omega$ to a convex, compact set
$\tilde{\Omega}\subset\mathbb{R}^{2}$. Then, $g(t)$ in (112) is equal to the
function

$t\mapsto-\cos_{\tilde{\Omega}}(0)\sin_{\tilde{\Omega}}(kt).$ (113)

The conclusion immediately follows from Corollary 5.5. ∎

######  Corollary 5.7.

Given a non-null constant $k\in\mathbb{R}$ and every angle $\psi$, the
function

$h:\mathbb{R}\to\mathbb{R};\qquad
h(t):=1-\sin_{\Omega^{\circ}}(\psi)\sin_{\Omega}\big{(}(\psi+kt)_{\circ}\big{)}-\cos_{\Omega^{\circ}}(\psi)\cos_{\Omega}\big{(}(\psi+kt)_{\circ}\big{)}.$
(114)

is non-decreasing for positive values of $t$ and non-increasing for negative
values of $t$, locally around $0$.

###### Proof.

Note that $h$ is the derivative of the function

$\mathbb{R}\ni t\mapsto
kt+\sin_{\Omega^{\circ}}(\psi)\cos_{\Omega^{\circ}}(\psi+kt)-\cos_{\Omega^{\circ}}(\psi)\sin_{\Omega^{\circ}}(\psi+kt),$
(115)

divided by $k$. The thesis follows from Corollary 5.6, since the function
(115) is the sum of a linear function and of a function of the type (112). ∎

In the following we are going to consider the trigonometric functions
associated to the closed unit ball of a strictly convex norm
${\|\\!\cdot\\!\|}$ on $\mathbb{R}^{2}$, i.e.
$\Omega:=\overline{B}^{\left\lVert\cdot\right\rVert}_{1}(0)$. In this case,
the polar set $\Omega^{\circ}$ is the closed unit ball
$\overline{B}^{\left\lVert\cdot\right\rVert_{*}}_{1}(0)$ of the dual norm
${\|\\!\cdot\\!\|}_{*}$. Moreover, according to the Pythagorean identity
(111), if $\theta\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega}\psi$
${\|\\!\cdot\\!\|}$ then $Q_{\psi}$ is a dual vector of $P_{\theta}$. In
particular, if ${\|\\!\cdot\\!\|}$ is a $C^{1}$ norm, Lemma 2.9 ensures that

$(\cos_{\Omega^{\circ}}(\psi),\sin_{\Omega^{\circ}}(\psi))=Q_{\psi}=d_{P_{\theta}}{\|\\!\cdot\\!\|}=d_{(\cos_{\Omega}(\theta),\sin_{\Omega}(\theta))}{\|\\!\cdot\\!\|}.$
(116)

### 5.2 Geodesics in the Heisenberg group

We present here the sub-Finsler Heisenberg group and study its geodesics. Let
us consider the Lie group $M=\mathbb{R}^{3}$, equipped with the non-
commutative group law, defined by

$(x,y,z)\star(x^{\prime},y^{\prime},z^{\prime})=\bigg{(}x+x^{\prime},y+y^{\prime},z+z^{\prime}+\frac{1}{2}(xy^{\prime}-x^{\prime}y)\bigg{)},\qquad\forall\,(x,y,z),(x^{\prime},y^{\prime},z^{\prime})\in\mathbb{R}^{3},$
(117)

with identity element ${\rm e}=(0,0,0)$. In the notation of Section 2.3, we
define the following morphism of bundles

$\xi:M\times\mathbb{R}^{2}\to
TM,\qquad\xi(x,y,z;u_{1},u_{2})=\bigg{(}x,y,z;u_{1},u_{2},\frac{1}{2}(u_{2}x-u_{1}y)\bigg{)}.$
(118)

The associated distribution of rank $2$ is spanned by the following left-
invariant vector fields:

$X_{1}=\partial_{x}-\frac{y}{2}\partial_{z},\qquad
X_{2}=\partial_{y}+\frac{x}{2}\partial_{z},$ (119)

namely $\mathcal{D}=\text{span}\\{X_{1},X_{2}\\}$. It can be easily seen that
$\mathcal{D}$ is bracket-generating. Then, letting
${\|\\!\cdot\\!\|}:\mathbb{R}^{2}\to\mathbb{R}_{+}$ be a norm, the _sub-
Finsler Heisenberg group_ $\mathbb{H}$ is the Lie group $M$ equipped with the
sub-Finsler structure $(\xi,{\|\\!\cdot\\!\|})$. By construction, also the
resulting norm on the distribution is left-invariant, so that the left-
translations defined by

$L_{p}:\mathbb{H}\to\mathbb{H};\qquad L_{p}(q):=p\star q,$ (120)

are isometries for every $p\in\mathbb{H}$.

In this setting, the geodesics were originally studied in [Bus47] and [Ber94]
for the three-dimensional case and in [Lok21] for general left-invariant
structures on higher-dimensional Heisenberg groups. We recall below the main
results of [Ber94] for strictly convex norms.

######  Theorem 5.8 ([Ber94, Thm. 1, Thm. 2]).

Let $\mathbb{H}$ be the sub-Finsler Heisenberg group, equipped with a strictly
convex norm. Then, the following statements hold:

  1. (i)

for any $q\in\mathbb{H}\setminus\\{x=y=0\\}$, there exists a unique geodesic
$\gamma:[0,1]\to\mathbb{H}$ joining the origin and $q$.

  2. (ii)

$\gamma:[0,T]\to\mathbb{H}$ is a geodesic starting at the origin if and only
if it satisfies the Pontryagin’s maximum principle for the time-optimal
control problem:

$\begin{cases}\dot{\gamma}(t)=\displaystyle
u_{1}(t)X_{1}(\gamma(t))+u_{2}(t)X_{2}(\gamma(t)),\\\ \displaystyle
u(t)\in\overline{B}^{\left\lVert\cdot\right\rVert}_{1}(0),\quad\gamma(0)=q_{0},\quad\text{and}\quad\gamma(T)=q_{1},\\\
T\to\min.\end{cases}$ (121)

######  Remark 5.9.

Note that the geodesics in [Ber94] are found solving the Pontryagin maximum
principle of Theorem 3.2 for the time-optimal problem. The latter is an
equivalent formulation of (P), however it produces arc-length parameterized
geodesics.

The next step is to compute explicitly the exponential map. In [Lok21], the
author provides an explicit expression for geodesics starting at the origin,
using the convex trigonometric functions functions presented in Section 5.1.
Since therein the author solves the time-optimal problem, we prefer to solve
explicitly the Hamiltonian system (H), in the case of the sub-Finsler
Heisenberg group.

######  Proposition 5.10 ([Lok21, Thm. 4]).

Let $\mathbb{H}$ be the sub-Finsler Heisenberg group, equipped with a strictly
convex norm. Let $\gamma:[0,1]\to\mathbb{H}$ be the projection of a (non-
trivial) normal extremal $(\lambda_{t})_{t\in[0,1]}$ starting at the origin,
then $\gamma(t)=(x(t),y(t),z(t))$, with

$\begin{cases}\begin{aligned}
x(t)&=\frac{r}{\omega}\left(\sin_{\Omega^{\circ}}(\phi+\omega
t)-\sin_{\Omega^{\circ}}(\phi)\right),\\\
y(t)&=-\frac{r}{\omega}\left(\cos_{\Omega^{\circ}}(\phi+\omega
t)-\cos_{\Omega^{\circ}}(\phi)\right),\\\
z(t)&=\frac{r^{2}}{2\omega^{2}}\left(\omega
t+\cos_{\Omega^{\circ}}(\phi+\omega
t)\sin_{\Omega^{\circ}}(\phi)-\sin_{\Omega^{\circ}}(\phi+\omega
t)\cos_{\Omega^{\circ}}(\phi)\right),\end{aligned}\end{cases}$ (122)

for some $\phi\in[0,2\mathbb{S}^{\circ})$,
$\omega\in\mathbb{R}\setminus\\{0\\}$ and $r>0$. If $\omega=0$, then

$\begin{cases}\begin{aligned}
x(t)&=\left(r\cos_{\Omega}(\phi_{\circ})\right)t,\\\
y(t)&=\left(r\sin_{\Omega}(\phi_{\circ})\right)t,\\\
z(t)&=0.\end{aligned}\end{cases}$ (123)

###### Proof.

Firstly, we characterize the sub-Finsler Hamiltonian in the sub-Finsler
Heisenberg group. Note that, without assuming additional regularity on
${\|\\!\cdot\\!\|}$, we can not apply directly Lemma 3.6. Nevertheless, we can
still obtain an analogous result by means of convex trigonometry. Indeed, let
$h_{i}(\lambda):=\langle\lambda,X_{i}(\pi(\lambda))\rangle$ for $i=1,2$, then

$H(\lambda):=\max_{u\in\mathbb{R}^{2}}\left(\sum_{i=1}^{2}u_{i}h_{i}(\lambda)-\frac{\left\lVert
u\right\rVert}{2}\right),\qquad\forall\,\lambda\in\ T^{*}\mathbb{H}.$ (124)

We introduce polar coordinates on $\mathbb{R}^{2}$ associated with
${\|\\!\cdot\\!\|}$ and its dual norm ${\|\\!\cdot\\!\|}_{*}$, namely
$(u_{1},u_{2})\mapsto(\rho,\theta)$ and $(h_{1},h_{2})\mapsto(\zeta,\psi)$
where

$\begin{cases}u_{1}=\rho\cos_{\Omega^{\circ}}(\theta),\\\
u_{2}=\rho\sin_{\Omega^{\circ}}(\theta),\end{cases}\qquad\text{and}\qquad\begin{cases}h_{1}=\zeta\cos_{\Omega^{\circ}}(\psi),\\\
h_{2}=\zeta\sin_{\Omega^{\circ}}(\psi).\end{cases}$ (125)

Hence, the sub-Finsler Hamiltonian becomes

$\begin{split}H(\lambda)&=\max_{u\in\mathbb{R}^{2}}\left(\sum_{i=1}^{2}u_{i}h_{i}(\lambda)-\frac{\left\lVert
u\right\rVert}{2}\right)\\\
&=\max_{\begin{subarray}{c}\theta\in[0,2\mathbb{S})\\\
\rho>0\end{subarray}}\left(\rho\zeta\left(\cos_{\Omega}(\theta)\cos_{\Omega^{\circ}}(\psi)+\sin_{\Omega}(\theta)\sin_{\Omega^{\circ}}(\psi)\right)-\frac{\rho^{2}}{2}\right)\leq\max_{\rho>0}\left(\rho\zeta-\frac{\rho^{2}}{2}\right)=\frac{\zeta^{2}}{2},\end{split}$
(126)

where the last inequality is a consequence of (110). Moreover, we attain the
equality in (126) if and only if $\rho=\zeta$ and $\psi=C^{\circ}(\theta)$.
Therefore, since $\zeta=\|\hat{\lambda}\|_{*}$ with
$\hat{\lambda}=(h_{1}(\lambda),h_{2}(\lambda))$, we conclude that

$H(\lambda)=\frac{1}{2}\big{\|}\hat{\lambda}\big{\|}^{2}_{*},\qquad\forall\,\lambda\in
T^{*}\mathbb{H}\setminus\operatorname{Ann}(\mathcal{D}),$ (127)

and the maximum is attained at the control $u=\hat{\lambda}^{*}$. Furthermore,
$H\in C^{1}(T^{*}M)$ by strict convexity of ${\|\\!\cdot\\!\|}$, cf.
Proposition 2.8. We write the system (51) in coordinates
$(x,y,z;h_{1},h_{2},h_{3})$ for the cotangent bundle, where
$h_{3}(\lambda):=\langle\lambda,\partial_{z}\rangle$. The vertical part of
(51) becomes

$\begin{cases}\displaystyle\dot{h}_{1}(t)=\|\hat{\lambda}_{t}\|_{*}d_{\hat{\lambda}_{t}}{\|\\!\cdot\\!\|}_{*}\cdot\left(0,-h_{3}(t)\right),\\\
\displaystyle\dot{h}_{2}(t)=\|\hat{\lambda}_{t}\|_{*}d_{\hat{\lambda}_{t}}{\|\\!\cdot\\!\|}_{*}\cdot\left(h_{3}(t),0\right),\\\
\displaystyle\dot{h}_{3}(t)=0.\end{cases}$ (128)

Let $(\lambda_{t})_{t\in[0,1]}$ be a normal extremal with associated maximal
control given by $t\mapsto u(t)$, then we use Lemma 2.9 to deduce that
$\|\hat{\lambda}_{t}\|_{*}d_{\hat{\lambda}_{t}}{\|\\!\cdot\\!\|}_{*}=\hat{\lambda}_{t}^{*}=u(t)$.
Therefore, letting $h_{3}(t)\equiv\omega\in\mathbb{R}$, we may rewrite (128)
as

$\begin{cases}\displaystyle\dot{h}_{1}(t)=-\omega\,u_{2}(t),\\\
\displaystyle\dot{h}_{2}(t)=\omega\,u_{1}(t).\end{cases}$ (129)

To solve this system, we use the polar coordinates (125): letting