# Unified synthetic Ricci curvature lower bounds for Riemannian and sub-
Riemannian structures

Davide Barilari Barilari: Dipartimento di Matematica “Tullio Levi-Civita”,
Università di Padova, Via Trieste 63, Padova, Italy. E-mail:
<EMAIL_ADDRESS>Andrea Mondino Mondino: Mathematical Institute,
University of Oxford (UK). E-mail<EMAIL_ADDRESS>Luca Rizzi
Rizzi: SISSA, via Bonomea 265, 34136 Trieste, Italy and Univ. Grenoble Alpes,
CNRS, Institut Fourier, F-38000 Grenoble, France. E-mail<EMAIL_ADDRESS>
###### Abstract

Recent advances in the theory of metric measures spaces on the one hand, and
of sub-Riemannian ones on the other hand, suggest the possibility of a “great
unification” of Riemannian and sub-Riemannian geometries in a comprehensive
framework of synthetic Ricci curvature lower bounds, as put forth in [85, Sec.
9]. With the aim of achieving such a unification program, in this paper we
initiate the study of gauge metric measure spaces.

###### Contents

  1. 1 Introduction
  2. 2 Table of notations
  3. 3 Synthetic Ricci curvature lower bounds for gauge spaces
    1. 3.1 Preliminaries and notation
      1. 3.1.1 Convergence of measures
      2. 3.1.2 Optimal transport and $W_{2}$-distance in metric spaces
      3. 3.1.3 Relative entropies
    2. 3.2 Gauge functions on metric spaces
    3. 3.3 Curvature-dimension conditions for gauge spaces
      1. 3.3.1 Recovering the Lott-Sturm-Villani’s theory
      2. 3.3.2 Recovering the Balogh-Kristàly-Sipos’ theory
  4. 4 Geometric consequences
    1. 4.1 Generalized Brunn-Minkowski inequality
    2. 4.2 MCP implies CD for spaces supporting interpolation inequalities
    3. 4.3 Gauge diameter estimate
    4. 4.4 Doubling inequalities for balls
    5. 4.5 Geodesic dimension estimates
    6. 4.6 Generalized Bishop-Gromov inequality
    7. 4.7 Parametric doubling for gauge balls
  5. 5 Stability and compactness
    1. 5.1 Stability of $\mathsf{MCP}(\beta)$
    2. 5.2 Stability of $\mathsf{CD}(\beta,n)$
    3. 5.3 Compactness for $\mathsf{MCP}(\beta)$ and $\mathsf{CD}(\beta,n)$ spaces
  6. 6 Vector-valued gauge functions
    1. 6.1 Geometric consequences
    2. 6.2 Stability and compactness
  7. 7 Natural gauge functions
    1. 7.1 The $\mathsf{D}$ function
    2. 7.2 Definition of natural gauge functions
    3. 7.3 Natural gauge functions in sub-Riemannian metric spaces
      1. 7.3.1 Meek condition
      2. 7.3.2 Regularity condition
      3. 7.3.3 G-boundedness condition
    4. 7.4 Expression of $\mathsf{D}$ out of the cut locus
  8. 8 Sub-Riemannian comparison theory
    1. 8.1 LQ optimal control problems
    2. 8.2 Constant curvature models
      1. 8.2.1 The Riemannian case
      2. 8.2.2 The Sasakian case
      3. 8.2.3 The two columns case
    3. 8.3 Main comparison result
  9. 9 Curvature estimates for fat sub-Riemannian structures
    1. 9.1 Fat curvature estimates
    2. 9.2 Preliminary setup
      1. 9.2.1 Fundamental computations
      2. 9.2.2 Partial dilations
      3. 9.2.3 Pseudo-homogeneous maps
    3. 9.3 Regularity of the canonical frame
    4. 9.4 Regularity of the canonical curvature maps
    5. 9.5 Proof of Theorem 9.2
    6. 9.6 Compact fat structures satisfy the $\mathsf{CD}(\beta,n)$ condition
  10. 10 Examples and applications
    1. 10.1 Heisenberg group
      1. 10.1.1 Application of the comparison theory to the Heisenberg group
    2. 10.2 Canonical variation
      1. 10.2.1 Sasakian foliations
    3. 10.3 Convergence to the tangent cone
      1. 10.3.1 Description of convergence to the tangent cone
      2. 10.3.2 Convergence of natural gauge functions to the tangent cone
    4. 10.4 Vector-valued gauge functions on three-dimensional structures
  11. A Sub-Riemannian geometry
  12. B Canonical curvature
    1. B.1 Curves in the Lagrange Grassmannian
      1. B.1.1 Reduced Young diagram
      2. B.1.2 Normal mappings
      3. B.1.3 Normal moving frames
      4. B.1.4 Canonical splittings and curvatures
    2. B.2 Jacobi curves
      1. B.2.1 Curves in the Lagrange Grassmannian point of view
      2. B.2.2 Canonical frame along the extremal point of view
      3. B.2.3 Smooth families of operators point of view
    3. B.3 Geodesic volume derivative
    4. B.4 The Riemannian case

##  1 Introduction

The subject of this work is the synthetic treatment of curvature lower bounds.
To illustrate our contribution, let us start by recalling the celebrated
theory of Alexandrov spaces. These are metric spaces $(X,\mathsf{d})$ where
curvature bounds are defined via comparison of geodesic triangles in $X$ with
corresponding ones in model surfaces of constant curvature. The key
observation is that Toponogov triangle comparison theorem gives a purely
metric characterization of lower bounds for the sectional curvature on smooth
Riemannian manifolds, which yields a synthetic definition for metric spaces.

A synthetic characterization of _Ricci_ curvature lower bounds needs an
additional structure, namely a reference measure $\mathfrak{m}$: following the
seminal works by Gromov [49], by Fukaya on spectral convergence [47], by
Cheeger-Colding on Ricci limit spaces [40], it emerged that a natural
framework for Ricci curvature lower bounds is the one of metric measure
spaces.

A key fact is that, on a smooth Riemannian manifold, Ricci curvature lower
bounds can be characterized in terms of suitable convexity inequalities of
entropy functionals in the Wasserstein space [42, 87], after [66, 73]. This
line of research culminated in the seminal work by Lott-Sturm-Villani in the
2000’s [64, 81, 82], introducing the synthetic $\mathsf{CD}(K,N)$ conditions
for metric measure spaces. The latter conditions depend on two parameters:
$K\in\mathbb{R}$ playing the role of a lower bound on the Ricci curvature, and
$N\geq 1$ playing the role of an upper bound on the dimension. Concretely, $K$
and $N$ enter into the theory via model _distortion coefficients_
$\beta_{K,N}$, encoding the effect of the Ricci curvature on the distortion of
the measure along geodesics. These coefficients have the form:

$(\beta_{K,N})_{t}(\theta):=t\cdot\frac{\sin\left(t\theta\sqrt{K/(N-1)}\right)^{N-1}}{\sin\left(\theta\sqrt{K/(N-1)}\right)^{N-1}},\qquad\forall\,t\in[0,1],$
(1.1)

for $0<K\theta^{2}<(N-1)\pi^{2}$ and with standard interpretation otherwise,
see Section 3.3.1.

A fundamental aspect of the $\mathsf{CD}(K,N)$ condition is that it enjoys
compactness and stability properties under pmGH convergence. We recall that
the class of $\mathsf{CD}(K,N)$ spaces includes Riemannian manifolds with
Ricci curvature bounded from below, as well as Finsler ones [72].

However, there is an important class of metric measure spaces which does not
fit into this framework: sub-Riemannian structures. These are smooth manifolds
endowed with a length metric obtained by considering only curves that are
tangent to a bracket-generating vector distribution (see Appendix A for a
self-contained account).

The simplest examples are the Heisenberg groups. In [53], Juillet proved that
the $\mathsf{CD}(K,N)$ condition fails for all values of $K\in\mathbb{R}$ and
$N\geq 1$ on the Heisenberg groups, and indeed this is the case also in more
general sub-Riemannian manifolds [15, 54, 65]. Moreover, in [53], it also
proved that a weaker synthetic condition, known as measure contraction
property $\mathsf{MCP}(K,N)$, holds in Heisenberg groups for suitable values
of $K$ and $N$. This property has been proved to hold in more general Carnot
groups [18, 74].

Recently, in their seminal work [19], Balogh-Kristály-Sipos showed that,
despite the failure of the classical $\mathsf{CD}(K,N)$ condition, weaker
entropy inequalities hold in the Heisenberg groups $\mathbb{H}^{d}$, with the
following distortion coefficients:

$\beta^{\mathbb{H}^{d}}_{t}(\theta):=t\cdot\dfrac{\sin\left(\frac{t\theta}{2}\right)^{2d-1}\left[\sin\left(\frac{t\theta}{2}\right)-\frac{t\theta}{2}\cos\left(\frac{t\theta}{2}\right)\right]}{\sin\left(\frac{\theta}{2}\right)^{2d-1}\left[\sin\left(\frac{\theta}{2}\right)-\frac{\theta}{2}\cos\left(\frac{\theta}{2}\right)\right]},\qquad\forall\,t\in[0,1],$
(1.2)

for $0<\theta<2\pi$, and standard interpretation otherwise. See also [20] for
similar inequalities for general co-rank $1$ Carnot groups.

Two remarkable differences between the Riemannian distortion coefficients
(1.1) and the Heisenberg ones (1.2) are:

  * •

as expected, the functional expressions (1.1) and (1.2) do not match;

  * •

more strikingly, when used to encode measure distortion, while in the
Riemannian coefficients (1.1), the parameter $\theta$ takes as value the
_length_ , in the Heisenberg ones (1.2), $\theta$ takes as value the
_curvature_ of geodesics (as curves in $\mathbb{R}^{2d+1}$).

A different approach to Ricci curvature lower bounds in sub-Riemannian
manifolds was put forward by Baudoin-Garofalo [27]. Inspired by the Bakry-
Émery semi-group techniques, they generalized curvature-dimension conditions
for sub-Riemannian structures with symmetries, introducing a suitable
generalization of Bochner’s identity and $\Gamma$-calculus. See the lecture
notes [26] and references therein for an account of this theory. We also
mention the recent work by Stefani [79] which, in the setting of Carnot
groups, establishes a first link between the Lagrangian approach to Ricci
curvature lower bounds (dealing with convexity-type properties of entropy
along Wasserstein geodesics) and the Eulerian one (focused instead on the
properties of the heat flow).

Motivated by the aforementioned contributions, in his 2017 Bourbaki Seminar,
Villani envisioned the possibility of a “great unification” of Riemannian and
sub-Riemannian geometries in a comprehensive theory of synthetic Ricci
curvature lower bounds. See [85, Sec. 9, conclusions et perspectives], and
[84, Rmk. 14.23]. Developing such a theory is the ambition of the present
paper.

### Gauge metric measure spaces

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space, m.m.s. for short.
We add a supplementary structure to the metric measure one, namely a non-
negative Borel function $\mathsf{G}:X\times X\to[0,+\infty]$, that we call
_gauge function_. The quadruple $(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ will
be called _gauge metric measure space_.

The following analogy explains the role of the gauge function. A Riemannian
manifold has Ricci curvature bounded from below by $K\in\mathbb{R}$ if for any
pair of points $x,y$ and any geodesic $\gamma$ between $x$ and $y$ it holds

$\mathrm{Ric}(\dot{\gamma},\dot{\gamma})\geq
K\|\dot{\gamma}\|^{2}=K\mathsf{d}(\gamma_{0},\gamma_{1})^{2}.$ (1.3)

The distance function $\mathsf{d}$ is used in the right hand side as a _gauge_
to measure the _extent_ of the lower Ricci curvature bound, quantified by the
constant $K$. The idea is to replace the distance $\mathsf{d}$ with a general
gauge function $\mathsf{G}$.

Gauge functions will be a key object in our extension of the synthetic theory
of Ricci curvature bounds to the sub-Riemannian setting, where it is now well-
understood that the effect of geometry on transport inequalities may not
depend uniquely on the distance, but rather on other intrinsic sub-Riemannian
quantities (see [19], [24, Sec. 8.1]).

### Distortion coefficients

Motivated by the comparison theory in sub-Riemannian geometry [8, 11, 89, 62,
51, 5, 61, 22, 25] (from the Hamiltonian viewpoint) and the discussion above,
we now introduce general distortion coefficients.

Let $\mathsf{s}:[0,+\infty)\to\mathbb{R}$ be a continuous function and
$N\in[1,+\infty)$ such that

$\mathsf{s}(\theta)=c\,\theta^{N}+o(\theta^{N})\qquad\text{ as $\theta\to
0$},$ (1.4)

for some $c>0$. The parameter $N$ will be a sharp upper bound for a new notion
of dimension, which is in general different from the Hausdorff one, see
Section 4.5. Denote:

$\mathcal{D}:=\inf\\{\theta>0\mid\mathsf{s}(\theta)=0\\}.$ (1.5)

It is clear that $\mathcal{D}>0$. The latter will be a sharp upper bound on
the gauge function, see Section 4.3. Define the _distortion coefficient_
$\beta_{(\cdot)}(\cdot):[0,1]\times[0,+\infty]\to[0,+\infty]$ as

$\displaystyle(t,\theta)\in[0,1]\times[0,+\infty]\mapsto\beta_{t}(\theta):=\begin{cases}t^{N}&\theta=0,\\\
\dfrac{\mathsf{s}(t\theta)}{\mathsf{s}(\theta)}&0<\theta<\mathcal{D},\\\
\displaystyle\liminf_{\phi\to\mathcal{D}^{-}}\dfrac{\mathsf{s}(t\phi)}{\mathsf{s}(\phi)}&\theta\geq\mathcal{D}.\end{cases}$
(1.6)

Notice that, when properly understood, both the Riemannian (1.1) and the
Heisenberg (1.2) distortion coefficients are obtained as in (1.6), for a
suitable $\mathsf{s}$.

In applications to sub-Riemannian geometry, the function $\mathsf{s}$ is
chosen in a class of models characterized as solutions to suitable ODEs. We
illustrate this characterization in Section 8, and in particular we refer to
Proposition 8.5 and Remark 8.6. Here we develop the theory in full generality,
without further constraints on the function $\mathsf{s}$.

### Entropy functionals

We consider the metric space $(\mathcal{P}_{2}(X),W_{2})$ of probability
measures with finite second moment endowed with the Kantorovich-Rubinstein-
Wasserstein quadratic transportation distance, see Section 3.1.2 for the
definitions. A $W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ can be equivalently
represented by a probability measure $\nu$ on the space of geodesics
$\mathrm{Geo}(X)$, with $\mu_{t}=({\rm e}_{t})_{\sharp}\nu$, where ${\rm
e}_{t}:\mathrm{Geo}(X)\to X$ is the evaluation map at time $t$. Such a $\nu$
is called _optimal dynamical plan_ from $\mu_{0}$ to $\mu_{1}$, and the set of
such measures is denoted by ${\rm OptGeo}(\mu_{0},\mu_{1})$. Let also
$\mathcal{P}_{ac}(X,\mathfrak{m})$ be the space of probability measures that
are absolutely continuous with respect to $\mathfrak{m}$.

Recall that, for $\mu\in\mathcal{P}_{2}(X)$, its relative (Boltzmann-Shannon)
entropy is defined by

$\mathrm{Ent}(\mu|\mathfrak{m}):=\int_{X}\rho\log\rho\,\mathfrak{m},\qquad\text{
if
$\mu=\rho\,\mathfrak{m}\in\mathcal{P}_{2}(X)\cap\mathcal{P}_{ac}(X,\mathfrak{m})$},$
(1.7)

in case $\rho\log\rho\in L^{1}(X,\mathfrak{m})$, otherwise we set
$\mathrm{Ent}(\mu|\mathfrak{m}):=+\infty$.

Let $\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))$ be the finiteness domain
of the entropy and

$\displaystyle\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$
$\displaystyle:=\\{\mu\in\mathcal{P}(X,\mathsf{d})\mid\text{
$\operatorname{\mathrm{supp}}\,\mu$ is bounded and
}\operatorname{\mathrm{supp}}\,\mu\subseteq\operatorname{\mathrm{supp}}\,\mathfrak{m}\\},$
(1.8) $\displaystyle\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$
$\displaystyle:=\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\cap\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m}).$
(1.9)

The subspaces $\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$ and
$\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ will play a key role
throughout the paper.

In order to formulate “dimensional” Ricci curvature lower bounds, it is
convenient to introduce also the following dimensional entropy (cf. [44]):

$\mathrm{U}_{n}(\mu|\mathfrak{m}):=\exp\left(-\frac{\mathrm{Ent}(\mu|\mathfrak{m})}{n}\right),\qquad
n\in[1,+\infty),$ (1.10)

with the understanding that $\mathrm{U}_{n}(\mu|\mathfrak{m}):=0$ if
$\mu\notin\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))$.

###  $\mathsf{CD}(\beta,n)$ spaces and $\mathsf{MCP}(\beta)$ spaces

We introduce synthetic Ricci curvature lower bounds on gauge m.m.s.: the
Curvature-Dimension condition $\mathsf{CD}(\beta,n)$ and the Measure
Contraction Property $\mathsf{MCP}(\beta)$.

######  Definition (Definition 3.5).

Let $n\in[1,+\infty)$, and $\beta$ as in (1.6). A gauge metric measure space
$(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies:

  * •

$\mathsf{CD}(\beta,n)$ if for all
$\mu_{0}\in\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$,
$\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$,
there exists a $W_{2}$-geodesic
$(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(X,\mathsf{d})$ connecting them,
induced by $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$, such that it holds

$\mathrm{U}_{n}(\mu_{t}|\mathfrak{m})\geq\exp\left(\frac{1}{n}\int_{\mathrm{Geo}(X)}\log\beta_{1-t}\big{(}\mathsf{G}(\gamma_{1},\gamma_{0}))\,\nu(\mathrm{d}\gamma)\right)\mathrm{U}_{n}(\mu_{0}|\mathfrak{m})\\\
+\exp\left(\frac{1}{n}\int_{\mathrm{Geo}(X)}\log\beta_{t}\big{(}\mathsf{G}(\gamma_{0},\gamma_{1})\big{)}\,\nu(\mathrm{d}\gamma)\right)\mathrm{U}_{n}(\mu_{1}|\mathfrak{m}),\qquad\forall\,t\in(0,1),$
(1.11)

with the convention that $\infty\cdot 0=0$.

  * •

$\mathsf{MCP}(\beta)$ if for any
$\bar{x}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ and
$\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\bar{x}\notin\operatorname{\mathrm{supp}}\mu_{1}$ there exists a
$W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(X,\mathsf{d})$
from $\mu_{0}=\delta_{\bar{x}}$ to $\mu_{1}$ such that

$\mathrm{U}_{n}(\mu_{t}|\mathfrak{m})\geq\exp\left(\frac{1}{n}\int_{X}\log\beta_{t}\big{(}\mathsf{G}(\bar{x},x)\big{)}\,\mu_{1}(\mathrm{d}x)\right)\mathrm{U}_{n}(\mu_{1}|\mathfrak{m}),\qquad\forall\,t\in(0,1),$
(1.12)

for some (and then every) $n\geq 1$.

We anticipate here a few remarks.

  1. 1.

The $\mathsf{MCP}(\beta)$ condition does not depend on the value of $n$. See
Remark 3.6.

  2. 2.

By construction, $\mathsf{CD}(\beta,n)\Rightarrow\mathsf{MCP}(\beta)$. See
Remark 3.7.

  3. 3.

Conversely, assuming that $(X,\mathsf{d},\mathfrak{m})$ supports interpolation
inequalities for densities with dimensional parameter $n$ (in the sense of
Definition 4.5), we show that
$\mathsf{MCP}(\beta)\Rightarrow\mathsf{CD}(\beta,n)$. See Theorem 4.7.

  4. 4.

In contrast with the Lott-Sturm-Villani’s theory where $\mu_{0},\mu_{1}$ are
both absolutely continuous, here non-absolutely continuous $\mu_{0}$ are
allowed. See Remark 3.8.

  5. 5.

The $\mathsf{CD}/\mathsf{MCP}$ conditions imply that
$\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\subset\mathcal{P}_{2}(X,\mathsf{d})$
(with metric $W_{2}$) and $\operatorname{\mathrm{supp}}\mathfrak{m}$ (with
metric $\mathsf{d}$) are length spaces. See Remark 3.9.

### Compatibility with classical synthetic theories

The $\mathsf{CD}(\beta,n)$ and $\mathsf{MCP}(\beta)$ conditions satisfy the
following compatibility properties:

  * •

_Compatibility with Lott-Sturm-Villani’s $\mathsf{CD}$_: by choosing as
distortion coefficients the Riemannian ones, i.e. $\beta=\beta_{K,N}$ as in
(1.1), and as gauge function the distance, i.e. $\mathsf{G}=\mathsf{d}$, we
show that, for essentially non-branching m.m.s., the Lott-Sturm-Villani’s
$\mathsf{CD}(K,N)$ conditions are equivalent to the corresponding
$\mathsf{CD}(\beta_{K,N},N)$. See Section 3.3.1.

  * •

_Compatibility with Ohta-Sturm’s $\mathsf{MCP}$_: as above, we show that the
Ohta-Sturm $\mathsf{MCP}(K,N)$ are equivalent to the corresponding
$\mathsf{MCP}(\beta_{K,N})$. See Section 3.3.1.

  * •

_Compatibility with Balogh-Kristály-Sipos_ : by choosing as distortion
coefficients the Heisenberg ones, i.e. $\beta=\beta^{\mathbb{H}^{d}}$ as in
(1.2), and as gauge function $\mathsf{G}(x,y)=\theta^{x,y}$ where $\theta$ is
the curvature of the geodesic from $x$ to $y$ (as a curve in
$\mathbb{R}^{2d+1}$), we show that the Heisenberg group $\mathbb{H}^{d}$
(endowed with the Carnot-Carathéodory distance, and the $2d+1$ Lebesgue
measure) satisfies the $\mathsf{CD}(\beta^{\mathbb{H}^{d}},2d+1)$ condition.
See Section 3.3.2.

  * •

_Compatibility with E. Milman’s $\mathsf{CGTD}$_: the Curvature-Geodesic-
Topological-Dimension conditions $\mathsf{CGTD}(K,N,n)$ introduced by E.
Milman in [68, Sec. 7], for $K\in\mathbb{R}$, $n\geq 1$ and $N\geq n$ is
equivalent to the $\mathsf{CD}(\beta_{K,N},n)$ condition, for gauge function
$\mathsf{G}=\mathsf{d}$. See Remark 3.14.

### Geometric consequences

One of the novel features of the $\mathsf{CD}(\beta,n)$ and
$\mathsf{MCP}(\beta)$ conditions above is the interplay of the gauge function
$\mathsf{G}$ (measuring the extent of the Ricci curvature lower bounds) and
the distance $\mathsf{d}$ (governing the geodesics and thus optimal
transport). This leads to a decoupling between metric vs distortion aspects in
the classical geometric consequences of the curvature-dimension conditions. In
Section 4 we will obtain the following results:

  * •

_Generalised Brunn-Minkowski inequality_ : given two sets $A_{0},A_{1}$, we
estimate from below the volume of the set $A_{t}$ of $t$-intermediate points
of geodesics from $A_{0}$ to $A_{1}$, with a distortion which is quantified
_purely in terms of the gauge function $\mathsf{G}$ and the general distortion
coefficients $\beta$_. See Section 4.1.

  * •

_$\mathsf{MCP}\Rightarrow\mathsf{CD}$_ for spaces supporting interpolation
inequalities. See Section 4.2.

  * •

_Gauge-diameter estimates_ : the parameter $\mathcal{D}$ in (1.5) yields an
upper bound on the essential supremum of the gauge function, called gauge
diameter. A remarkable difference with respect to classical Bonnet-Myers
Theorem is that the gauge diameter can be bounded also for non-compact metric
measure spaces (e.g. the Heisenberg group, where the estimate we obtain is
sharp) and, conversely, unbounded for compact ones. See Section 4.3.

  * •

_Local doubling inequalities for metric balls_ : in contrast with the
classical case when “locality” is measured in terms of the distance, here it
is expressed in terms of the gauge function which also determines the doubling
constant. See Section 4.4.

  * •

_Geodesic dimension estimates_ : the parameter $N$ occurring in (1.4) yields
an upper bound for the Hausdorff dimension and for a notion of dimension for
m.m.s. recently introduced in [3, 77], the so-called geodesic dimension. See
Section 4.5. The latter bound is sharp while the former is not, see Remark
4.20.

  * •

_Generalised Bishop-Gromov inequalities_ : we obtain volume estimates on the
(truncated) sub-level sets of the gauge function. Such sets are reminiscent of
the “butterfly-shaped” sets appearing in [53, Fig. 2]. For the validity of the
Bishop-Gromov inequality, we need an additional compatibility condition
between the distance and the gauge function, which we name _meek property_.
See Section 4.6.

### Stability and compactness

Key features of the Lott-Sturm-Villani’s theory are the stability and
compactness properties with respect to the measured-Gromov-Hausdorff
convergence (mGH for short).

For a sequence of gauge m.m.s. it is natural to consider the mGH convergence
of the metric measure structures, coupled with an additional notion of
convergence for the gauge functions. A naive generalisation of the mGH
convergence to gauge functions, see (5.1), would fail for natural sequences
(e.g. convergence to the tangent cone for sub-Riemannian structures, cf.
Section 10.3). This is due to the low regularity of the gauge functions which
in general are not even continuous, in sharp contrast with the Lott-Sturm-
Villani’s theory where $\mathsf{G}=\mathsf{d}$ is clearly $1$-Lipschitz.

For this reason, in Section 5, we introduce a significantly weaker condition
which provides a good balance yielding both _stability_ and _compactness_
properties for the $\mathsf{CD}(\beta,n)$ condition and the
$\mathsf{MCP}(\beta)$. We ask for a sort of $L^{1}_{\mathrm{loc}}$ convergence
of gauge functions in varying spaces, and a regularity condition on the limit
gauge function outside of the diagonal. Compared to the classical Lott-Sturm-
Villani’s theory, the low regularity of the gauge functions and their weaker
convergence introduce new challenges in the proof.

### The vector-valued case

In Section 6 we extend the above theory to vector-valued gauge functions,
namely

$\mathsf{G}:X\times X\to\mathbb{RP}^{m}_{+},$ (1.13)

where $\mathbb{RP}^{m}_{+}$ is a projective semi-space. This generalization is
necessary in order to obtain sharp estimates for important classes of sub-
Riemannian examples, see Section 10.4.

### Natural gauge functions

In the Heisenberg group a natural gauge function is the quantity $\theta$
mentioned earlier in the introduction ($\theta^{x,y}$ is the curvature of the
geodesic from $x$ to $y$ as a curve of $\mathbb{R}^{2d+1}$). A key observation
is that $\theta$ can be written in terms of the Carnot-Carathéodory distance
and the norm of its gradient, computed with respect to a Riemannian extension.

This observation suggests a synthetic way to define natural gauge functions
for any metric measure space $(X,\mathsf{d},\mathfrak{m})$ equipped with a
reference metric $\mathsf{d}_{R}$ (the subscript stands for _Reference_ , or
_Riemannian_). The construction is well-suited to sub-Riemannian geometry
where, in several cases, one has natural Riemannian extensions serving as
reference.

Natural gauge functions will be constructed out of two main building blocks:
the distance function and the _$\mathsf{D}$ function_ defined as follows

$\mathsf{D}:X\times X\to[0,+\infty],\qquad\mathsf{D}(x,y):=\limsup_{z,w\to
x}\frac{|\mathsf{d}^{2}(z,y)-\mathsf{d}^{2}(w,y)|}{2\mathsf{d}_{R}(z,w)},$
(1.14)

with the convention that the limit is $0$ if $x$ is an isolated point.

Let us show how $\theta$ is obtained via the above procedure, in the
Heisenberg group $(\mathbb{H}^{d},\mathsf{d}_{cc},\mathscr{L}^{2d+1})$. As
extension $\mathsf{d}_{R}$, we choose the left-invariant Riemannian metric
obtained declaring the Reeb field $\partial_{z}$ to be of unit length and
orthogonal to the Heisenberg distribution. For $(x,y)$ out of the cut locus,
we prove in Proposition 7.28 that

$\theta^{x,y}=\sqrt{\mathsf{D}^{2}(x,y)-\mathsf{d}_{cc}^{2}(x,y)}.$ (1.15)

We note that $\mathsf{D}\geq\mathsf{d}$, see Proposition 7.5(i). Natural gauge
functions are studied in Section 7, in particular:

  * •

in Section 7.1 we define the $\mathsf{D}$ function and we establish its basic
properties;

  * •

in Section 7.2 we define _natural gauge functions_ on metric measure spaces
equipped with a reference metric, of which $\mathsf{D}$ and $\mathsf{d}$ are
particular cases;

  * •

in Section 7.3 we establish further properties of natural gauge functions,
namely:

    * –

the meek condition used for the generalized Bishop-Gromov Theorem; see Section
7.3.1;

    * –

the regularity and boundedness properties required for the stability and
compactness results; see Sections 7.3.2 and 7.3.3 respectively.

We prove that all these properties follow from natural hypotheses in sub-
Riemannian geometry, including: step $\leq$ 2, minimising Sard property,
$*$-minimizing Sard property, ideal, absence of non-trivial Goh geodesics,
real-analyticity. For a glimpse of all the implications see Figure 2;

  * •

in Section 7.4 we show how the natural gauge functions are related to the sub-
Riemannian distance out of the cut locus.

### Compatibility with the Hamiltonian theory of curvature

In Section 8 we establish the compatibility of the synthetic
$\mathsf{CD}(\beta,n)$ condition with the Hamiltonian theory of Ricci
curvature lower bounds for sub-Riemannian manifolds. The latter has its roots
in the pioneering works by Agrachev and Gamkrelidze on the geometry of curves
in Lagrange Grassmannians [8, 11], and was subsequently developed by Zelenko
and Li [89] with the introduction of the so-called _canonical frame_. This
technical tool, generalizing in a very broad sense the notion of parallel
transport, has been pivotal in the development of the comparison theory for
sub-Riemannian structures from the Hamiltonian point of view, started in [62,
5, 61]. Sub-Riemannian Ricci curvatures were finally introduced in [22] in
full generality, as _partial traces_ of the canonical curvatures. The
corresponding comparison theory for distortion coefficients was finally set in
[25].

We illustrate such a theory in Section 8, in a form adapted to our purposes.
For the sake of self-consistency, in Appendix B, we include an account of the
Agrachev-Zelenko-Li theory, that is used extensively in Sections 8-9.

We depict here the main ideas. Let $(M,\mathsf{d},\mathfrak{m})$ be a sub-
Riemannian metric measure space, i.e. $M$ is a smooth manifold, $\mathsf{d}$
is a sub-Riemannian distance, and $\mathfrak{m}$ is a smooth measure, see
Definition 7.9. The paradigm of sub-Riemannian comparison is that, for the
generic geodesic $\gamma$, one can decompose the tangent space along $\gamma$
in a family of subspaces, depending on the Lie bracket structure. These are
labeled by the boxes $\alpha$ of a Young diagram. For each subspace we define
a _canonical Ricci curvature_ $\mathfrak{Ric}^{\alpha}_{\gamma}$ as the
corresponding partial trace of the canonical curvature. We remark that in the
Riemannian case there is only one such a box $\alpha$, and

$\mathfrak{Ric}^{\alpha}_{\gamma}=\mathrm{Ric}(\dot{\gamma},\dot{\gamma}),$
(1.16)

recovering the classical Ricci tensor along that geodesic.

For a general sub-Riemannian metric measure space
$(M,\mathsf{d},\mathfrak{m})$, consider the “true” distortion coefficient of
the m.m.s.:

$\beta_{t}^{(M,\mathsf{d},\mathfrak{m})}(x,y):=\limsup_{r\to
0^{+}}\frac{\mathfrak{m}(Z_{t}(\\{x\\},B_{r}(y)))}{\mathfrak{m}(B_{r}(y))},\qquad\forall\,x,y\in
M,\quad t\in[0,1],$ (1.17)

see Definition 4.6. For fixed $(x,y)$ out of the cut locus, there is a unique
geodesic $\gamma$ joining $x$ with $y$, and when _every_ canonical Ricci
curvature along $\gamma$ is bounded from below,
$\mathfrak{Ric}^{\alpha}_{\gamma}\geq\kappa_{\alpha}$, for some
$\kappa_{\alpha}\in\mathbb{R}$, and all $\alpha$, then the following
comparison holds:

$\beta_{t}^{(M,\mathsf{d},\mathfrak{m})}(x,y)\geq\beta_{t}^{\mathrm{mod}},\qquad\forall\,t\in[0,1].$
(1.18)

Here, $\beta^{\mathrm{mod}}$ is a _model_ distortion coefficient. We
illustrate its properties.

  * •

$\beta^{\mathrm{mod}}$ is associated with a variational problem on
$\mathbb{R}^{n}$, of Linear-Quadratic type, coming from optimal control
theory. This is a change of paradigm with respect to classical Riemannian
comparison theory, where model distortion coefficients are the ones of
Riemannian space forms, while the models in the present theory are not even
metric spaces. See Section 8.1.

  * •

$\beta^{\mathrm{mod}}$ can be computed explicitly by solving a Hamiltonian
ODE, which is identified by the values of the Ricci lower bounds
$\kappa_{\alpha}$, and the structure of the Young diagram associated with the
geodesic $\gamma$. See Section 8.2.

  * •

$\beta^{\mathrm{mod}}$ has the form (1.6), in other words it is a model
distortion coefficient in the sense of the synthetic theory. This is not by
chance, and in fact Proposition 8.5, in particular item vi, is the bridge
between the curvature-dimension theory of Section 3 and the comparison theory
of Section 8.

  * •

When the Young diagram of $\gamma$ is of Riemannian type (i.e. it consists of
only one box), $\beta^{\mathrm{mod}}$ recovers the familiar Riemannian
coefficients. See Section 8.2.1.

  * •

The general class of $\beta^{\mathrm{mod}}$ contains _all_ distortion
coefficients observed so far in sub-Riemannian geometry: Heisenberg groups
[19], corank $1$ Carnot groups [20], Sasakian [5, 61] and $3$-Sasakian
manifolds [78]. See Sections 8.2.2 and 8.2.3.

The comparison result in the sense of (1.18) is Theorem 8.9, describing the
distortion along a single geodesic. If the Ricci curvature bounds hold
uniformly, in the precise technical sense of Definition 8.11, then one obtains
Theorem 8.12.

######  Theorem (Ideal sub-Riemannian structures with Ricci bounded below are
$\mathsf{CD}$).

Let $(M,\mathsf{d},\mathfrak{m})$ be an ideal sub-Riemannian metric measure
space, with $n=\dim M$, equipped with a finite gauge function
$\mathsf{G}:M\times M\to\mathbb{R}^{m}_{+}$, $m\in\mathbb{N}$, with Ricci
curvatures bounded from below in the sense of Definition 8.11, and let $\beta$
be the corresponding distortion coefficient. Then
$(M,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies the $\mathsf{CD}(\beta,n)$
condition.

This result establishes the compatibility of the $\mathsf{CD}(\beta,n)$ theory
with the one of sub-Riemannian Ricci bounds.

### Compact fat structures satisfy the curvature-dimension condition

Clearly, any compact Riemannian manifold has Ricci curvature bounded below,
and thus it verifies the $\mathsf{CD}(\beta,n)$ condition for suitable $\beta$
(of course, in this case, $\beta=\beta_{\kappa,n}$ where $\kappa$ is a lower
bound for the Ricci curvature and $n$ is the topological dimension). In
Section 9 we study the sub-Riemannian counterpart of such a statement, for the
case of fat distributions (also called strong bracket generating, see [70,
80]). We refer to (9.1) for the definition.

In order to apply Theorem 8.12, one must prove uniform lower bounds on the
Ricci curvatures of fat structures. In contrast with the Riemannian case, this
requires a considerable effort and delicate estimates, which are the object of
Section 9. There, we prove that, when equipped with the natural gauge function
$\mathsf{D}$ of (1.14), compact fat sub-Riemnnian metric measure spaces
satisfy the $\mathsf{CD}(\beta,n)$ condition, for an explicit $\beta$. See
Theorem 9.23 for a precise statement, which we anticipate here in a simplified
form.

######  Theorem (Compact fat structures are $\mathsf{CD}$).

Let $(M,\mathsf{d},\mathfrak{m})$ be a compact, $n$-dimensional, fat sub-
Riemannian metric measure space. Then for the natural gauge function
$\mathsf{D}:M\times M\to[0,+\infty)$ of (1.14), there exists an explicit
distortion coefficient $\beta$ such that
$(M,\mathsf{d},\mathfrak{m},\mathsf{D})$ satisfies the $\mathsf{CD}(\beta,n)$
condition.

As a direct consequence of this result, we obtain the following, see Corollary
9.24.

######  Corollary (Classical $\mathsf{MCP}$ for compact fat structures).

Let $(M,\mathsf{d},\mathfrak{m})$ be a compact $n$-dimensional sub-Riemannian
metric measure space, with fat distribution of rank $k<n$. Then there exists
$N^{\prime}\geq 3n-2k$ such that $(M,\mathsf{d},\mathfrak{m})$ satisfies the
classical $\mathsf{MCP}(0,N^{\prime})$.

The above result removes the real-analytic assumption of the analogous
statement by Badreddine and Rifford in [18, Thm. 1.3], in the case of fat
distributions, obtaining it with a completely different strategy.

### Examples and applications

Finally, in Section 10 we present examples and applications of the synthetic
theory to sub-Riemannian spaces.

  * •

In Section 10.1 we illustrate in detail our constructions in the case of the
three-dimensional Heisenberg group $\mathbb{H}^{1}$, including the computation
of the model distortion coefficients via the comparison theory of Section 8.

  * •

In Section 10.2 we study a one-parameter family of Riemannian metric measure
spaces, namely the canonical variations of the first Heisenberg group
$(\mathbb{H}^{1},\mathsf{d}_{\varepsilon},\mathscr{L}^{3})$. We discuss in
particular the sub-Riemannian limit ($\varepsilon\to 0$) and the adiabatic
limit ($\varepsilon\to+\infty$), both occurring within a single
$\mathsf{CD}(\beta,n)$ class.

  * •

In Section 10.3 we study the convergence to the tangent cone of sub-Riemannian
structures with natural gauge functions, and how the $\mathsf{CD}(\beta,n)$
condition behaves under the blow-up process.

  * •

In Section 10.4 we illustrate the vector-valued theory of Section 6, in the
setting of the three-dimensional left-invariant structures on Lie groups.

##### Acknowledgments.

Davide Barilari acknowledges support by the STARS Consolidator Grant 2021
“NewSRG” of the University of Padova. Andrea Mondino and Luca Rizzi have
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreements No.
802689 “CURVATURE” and No. 945655 “GEOSUB”).

##  2 Table of notations

Metric notation |
---|---
$(X,\mathsf{d},\mathfrak{m})$ |  metric measure space
$\bar{\mathfrak{m}}$ |  completion of $\mathfrak{m}$
$\mathrm{Geo}(X)$ |  set of length-minimizing, constant-speed curves $\gamma:[0,1]\to X$ of a metric space $(X,\mathsf{d})$
$B_{r}(x)$ |  open metric ball of radius $r$ and center $x$
$\dim_{\mathrm{Haus}}$ |  Hausdorff dimension
$\dim_{\mathrm{geo}}$ |  geodesic dimension, Definition 4.17
$W_{2}(\mu,\nu)$ |  $2$-Wasserstein distance, Section 3.1.2
${\rm e}_{t}$ |  evaluation maps $\gamma\mapsto\gamma_{t}$ at time $t$, Section 3.1.2
$\mathrm{Opt}(\mu,\nu)$ |  optimal couplings from $\mu$ to $\nu$, Section 3.1.2
$\mathrm{OptGeo}(\mu,\nu)$ |  optimal dynamical plans from $\mu$ to $\nu$, Section 3.1.2
$\mathrm{Ent}(\mu|\mathfrak{m})$ |  Boltzmann-Shannon entropy of $\mu$ w.r.t. $\mathfrak{m}$, Section 3.1.3
$\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))$ |  finiteness domain of $\mathrm{Ent}(\cdot|\mathfrak{m})$, Section 3.1.3
$\mathrm{U}_{n}(\mu|\mathfrak{m})$ |  $n$-dimensional entropy of $\mu$ w.r.t. $\mathfrak{m}$, Section 3.1.3
$\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$ |  probability measures with bounded support in $\operatorname{\mathrm{supp}}\mathfrak{m}$, Section 3.1.3
$\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ |  $\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\cap\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$, Section 3.1.3
$o(\cdot)$ |  function $o(\cdot):[0,+\infty)\to\mathbb{R}$ satisfying $\displaystyle\limsup_{\theta\downarrow 0}\frac{|o(\theta)|}{\theta}=0$
$\mathsf{G}$ |  gauge function, Eq. (3.10) scalar case, Eq. (6.2) vector case
$\mathsf{s}$ |  defining function for distortion coefficients, Eq. (3.12) scalar case, Eq. (6.3) vector case
$N$ |  order of $\mathsf{s}$ at 0, Eq. (3.12) scalar case, Eq. (6.3) vector case
$\mathcal{D}$ |  first zero of $\mathsf{s}$ in the scalar case, Eq. (3.13)
$\mathbb{RP}^{m}_{+}$ |  real projective semi-space, Section 6
${\rm{DOM}}$ |  positivity domain of $\mathsf{s}$ in the vector case, Section 6
$\mathcal{D}_{\theta}$ |  boundary of ${\rm{DOM}}$ along $\theta$, in the vector case, Section 6
$\beta$ |  distortion coefficient, scalar case Eq. (3.14), vector case Eq. (6.4)
$\beta^{\tau}_{K,N}$, $\beta^{\sigma}_{K,N}$ |  distortion coefficients for the Lott-Sturm-Villani’s theory, Section 3.3.1
$\beta^{\mathbb{H}^{d}}$ |  distortion coefficient for $\mathbb{H}^{d}$, Section 3.3.2
$\beta^{(X,\mathsf{d},\mathfrak{m})}$ |  distortion coefficient of a m.m.s., Definition 4.6
$\mathsf{MCP}(\beta)$ |  measure contraction property with distortion coefficient $\beta$, Definition 3.5
$\mathsf{CD}(\beta,n)$ |  curvature-dimension condition with distortion coefficient $\beta$ and dimensional parameter $n$, Definition 3.5
$A_{t},\,Z_{t}$ |  set of $t$-intermediate points, Eq. (4.1)
$\beta(A_{0},A_{1})$ |  distortion coefficient between the sets $A_{0},A_{1}$, Eq. (4.2)
$\operatorname{\mathrm{diam}}$ |  metric diameter
$\operatorname{\mathrm{diam}}_{\mathsf{G}}$ |  $\mathsf{G}$-diameter, Definition 4.8 scalar case, Definition 6.2 vector case
$\mathrm{v}_{\mathsf{G}}$, $\mathrm{s}_{\mathsf{G}}$ |  measures of “gauge balls” and “spheres”, Definition 4.21 scalar case, Definition 6.5 vector case
$\mathrm{Lip}^{\rho}_{a}$ |  asymptotic Lipschitz number w.r.t. a metric $\rho$, Definition 7.3
$\mathsf{D}$ |  $\mathsf{D}$ function, Definition 7.4
Sub-Riemannian notation |
$\langle\cdot,\cdot\rangle$ |  pairing of covectors with vectors
$\mathscr{D}$ |  distribution (possibly rank-varying), Appendix A
$\operatorname{\mathrm{End}}$ |  end-point map, Appendix A
$\mathscr{V}$ |  complement of a constant rank distribution, Section 7–9
$\mathrm{Cut}$ |  sub-Riemannian cut locus, Definition 7.11
$\beta^{A,B,Q}$ |  distortion coefficient of a general LQ problem, Definition 8.3
$\beta^{\kappa}$ or $\beta^{\bar{\kappa}}$ |  distortion coefficient of a model LQ problem, $\kappa\in\mathbb{R}^{\ell}$ Proposition 8.5iv, or $\bar{\kappa}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{\ell}$ Proposition 8.5vi. See Remark 8.7
$t_{\kappa}$ |  first conjugate time for model LQ problems, Proposition 8.5
${\rm{DOM}}_{\bar{\kappa}}$ |  positivity domain for model functions in the vector case, Proposition 8.5
$V^{0}$ |  annihilator of a vector space/bundle $V$, Section 9
$\Lambda_{\neq 0}$ |  for $\Lambda$, the set of $\lambda\in\Lambda\subseteq T^{*}M$ s.t. $H(\lambda)\neq 0$, Section 9.2
$\vec{H}$ |  Hamiltonian vector field, Appendix B
$Y$ |  reduced Young diagram of a Jacobi curve, Appendix B
$\Upsilon$ |  set of levels of a reduced Young diagram, Appendix B
$\mathrm{size}(\cdot)$ |  size of a superbox of a reduced Young diagram, Appendix B
$\\{E_{a},F_{b}\\}_{a,b\in Y}$ |  canonical frame, Appendix B
$\mathfrak{R}$ |  canonical curvature, Appendix B
$\mathfrak{Ric}$ |  canonical Ricci curvature, Appendix B
$\rho_{\mathfrak{m}}$ |  geodesic volume derivative, Appendix B
$\boxtimes,\boxplus,\boxdot,\boxminus$ |  superbox of a Young diagram (notation used in Section 9)

##  3 Synthetic Ricci curvature lower bounds for gauge spaces

###  3.1 Preliminaries and notation

####  3.1.1 Convergence of measures

Throughout all the paper, $(X,\mathsf{d})$ is a complete and separable metric
space. Denote with $\mathscr{M}_{\mathrm{loc}}(X)$ the set of Borel measures
with values in $[0,+\infty]$ which are finite on every bounded subset and with
$\mathscr{M}(X)$ the collection of all finite Borel measures. Notice that
every measure in $\mathscr{M}_{\mathrm{loc}}(X)$ is $\sigma$-finite, by the
exhaustion $X=\cup_{j\in\mathbb{N}}B_{j}(x)$, for some $x\in X$.

We endow $\mathscr{M}_{\mathrm{loc}}(X)$ with the (weak) topology induced by
the duality with the space $C_{bs}(X)$ of bounded and continuous function on
$X$ with bounded support: a sequence
$(\mu_{j})_{j\in\mathbb{N}}\subset\mathscr{M}_{\mathrm{loc}}(X)$ _converges
weakly_ to $\mu_{\infty}\in\mathscr{M}_{\mathrm{loc}}(X)$ if

$\lim_{j\to\infty}\int_{X}f\,\mu_{j}=\int_{X}f\,\mu_{\infty},\qquad\forall\,f\in
C_{bs}(X).$ (3.1)

Let $\mathcal{P}(X)$ denote the set of Borel probability measures and
$\mathcal{P}_{bs}(X,\mathsf{d})$ the set of probability measures with bounded
support.

When $(\mu_{j})_{j\in\mathbb{N}\cup\\{\infty\\}}\subset\mathcal{P}(X)$, the
weak convergence (3.1) is equivalent to the _narrow_ convergence, i.e.
convergence in $\mathscr{M}(X)$ in duality with the space $C_{b}(X)$ of
bounded and continuous function on $X$:

$\lim_{j\to\infty}\int_{X}f\,\mu_{j}=\int_{X}f\,\mu_{\infty},\qquad\forall\,f\in
C_{b}(X).$

Relative narrow compactness in $\mathcal{P}(X)$ can be characterised by
Prokhorov’s Theorem. In order to state it, recall that a subset
$\mathcal{K}\subset\mathcal{P}(X)$ is said to be _tight_ if, for every
$\varepsilon>0$, there exists a compact subset $K_{\varepsilon}\subset X$ such
that

$\mu(X\setminus
K_{\varepsilon})\leq\varepsilon,\qquad\forall\,\mu\in\mathcal{K}.$ (3.2)

######  Theorem 3.1 (Prokhorov).

Let $(X,\mathsf{d})$ be complete and separable, and let
$\mathcal{K}\subset\mathcal{P}(X)$. Then $\mathcal{K}$ is pre-compact in the
narrow topology if and only if $\mathcal{K}$ is tight.

####  3.1.2 Optimal transport and $W_{2}$-distance in metric spaces

Recall that if $f:X\to Y$ is a Borel map between the separable metric spaces
$(X,\mathsf{d}_{X})$, $(Y,\mathsf{d}_{Y})$, then any Borel (resp. probability)
measure $\mu$ can be _pushed forward_ to a Borel (resp. probability) measure
$f_{\sharp}\mu$ defined as $(f_{\sharp}\mu)(B):=\mu(f^{-1}(B))$ for every
Borel subset $B\subset Y$. Call $P_{i}:X\times X\to X$, $i=1,2$, the
projection on the $i^{th}$ factor. Given $\mu_{i}\in\mathcal{P}(X)$, $i=1,2$,
denote

$\mathrm{Cpl}(\mu_{1},\mu_{2}):=\\{\pi\in\mathcal{P}(X\times
X)\mid(P_{i})_{\sharp}\pi=\mu_{i},\,i=1,2\\}$

the set of _admissible couplings_ (also called “transference plans”) from
$\mu_{1}$ to $\mu_{2}$. Let $\mathcal{P}_{2}(X,\mathsf{d})$ the subspace of
probability measures with finite second moment, i.e.

$\mathcal{P}_{2}(X,\mathsf{d}):=\left\\{\mu\in\mathcal{P}(X)\mid\int_{X}\mathsf{d}(x,\bar{x})^{2}\,\mu(\mathrm{d}x)<\infty\text{
for some (and thus for all) }\bar{x}\in X\right\\}.$

We endow the space $\mathcal{P}_{2}(X)$ with the quadratic (Kantorovich-
Rubinstein-Wasserstein) transportation distance $W_{2}$ defined as

$W_{2}(\mu,\nu)^{2}:=\inf_{\pi\in\mathrm{Cpl}(\mu,\nu)}\int_{X\times
X}\mathsf{d}^{2}(x,y)\,\pi(\mathrm{d}x\mathrm{d}y).$

A coupling $\pi\in\mathrm{Cpl}(\mu,\nu)$ achieving the infimum in the right
hand side is called _optimal coupling_ from $\mu$ to $\nu$, and the set of
such optimal couplings is denoted by ${\rm Opt}(\mu,\nu)$. Since
$\mathrm{Cpl}(\mu,\nu)$ is non-empty and compact in the weak topology, it is
easily checked that ${\rm Opt}(\mu,\nu)\neq\emptyset$.

It is well-known that $(\mathcal{P}_{2}(X,\mathsf{d}),W_{2})$ is a complete
and separable metric space (see e.g. [84]). In order to discuss the relation
between narrow and $W_{2}$ convergence, recall that a subset
$\mathcal{K}\subset\mathcal{P}_{2}(X)$ is _$2$ -uniformly integrable_ provided
that

$\lim_{R\to\infty}\sup_{\mu\in\mathcal{K}}\int_{X\setminus
B_{R}(\bar{x})}\mathsf{d}^{2}(x,\bar{x})\,\mu(\mathrm{d}x)=0,\qquad\text{for
some (and thus for every) }\bar{x}\in X.$

For a proof of the following result see for instance [84, Thm. 6.8].

######  Proposition 3.2 (Characterization of $W_{2}$ convergence).

Let $(X,\mathsf{d})$ be a complete and separable metric space and
$(\mu_{j})_{j\in\bar{\mathbb{N}}}\subset\mathcal{P}_{2}(X,\mathsf{d})$. Then
the following are equivalent:

  * •

$(\mu_{j})_{j\in\mathbb{N}}$ is $2$-uniformly integrable and converges
narrowly to $\mu_{\infty}$;

  * •

$W_{2}(\mu_{j},\mu_{\infty})\to 0$ as $j\to\infty$.

We next recall some basics about the geodesic structure of
$(\mathcal{P}_{2}(X,\mathsf{d}),W_{2})$, cf. [84]. A _geodesic_ is a curve
$\gamma:[0,1]\to X$ satisfying

$\mathsf{d}(\gamma_{s},\gamma_{t})=|s-t|\,\mathsf{d}(\gamma_{0},\gamma_{1}),\qquad\forall\,s,t\in[0,1].$
(3.3)

In particular, geodesics are length-minimizing and parametrized with constant
speed on the interval $[0,1]$.

The space of all geodesics on $(X,\mathsf{d})$ is denoted by
$\mathrm{Geo}(X)$, which is endowed with the complete and separable distance

$\mathsf{d}_{\mathrm{Geo}(X)}(\gamma,\eta):=\sup_{t\in[0,1]}\mathsf{d}(\gamma_{t},\eta_{t}).$
(3.4)

Recall that $(X,\mathsf{d})$ is said to be a _geodesic space_ if every two
points in $X$ can be joined by a geodesic; $(X,\mathsf{d})$ is a geodesic
space if and only if $(\mathcal{P}_{2}(X,\mathsf{d}),W_{2})$ is so.

A useful procedure is to represent a geodesic in
$(\mathcal{P}_{2}(X,\mathsf{d}),W_{2})$ by a single probability measure
defined on $\mathrm{Geo}(X)$. More precisely: for every $W_{2}$-geodesic
$(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(X,\mathsf{d})$, there exists
$\nu\in\mathcal{P}(\mathrm{Geo}(X))$ such that $\mu_{t}=({\rm
e}_{t})_{\sharp}(\nu)$ for all $t\in[0,1]$, where

${\rm e}_{t}:\mathrm{Geo}(X)\to X,\qquad{\rm e}_{t}(\gamma):=\gamma_{t},$
(3.5)

is the evaluation map. Such a measure $\nu$ is called an _optimal dynamical
plan_ from $\mu_{0}$ to $\mu_{1}$, the set of which is denoted by ${\rm
OptGeo}(\mu_{0},\mu_{1})$. Given $\nu\in\mathcal{P}(\mathrm{Geo}(X))$, it
holds that $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ if and only if $({\rm
e}_{0},{\rm e}_{1})_{\sharp}\nu\in{\rm Opt}(\mu_{0},\mu_{1})$.

A set $\Gamma\subset\mathrm{Geo}(X)$ is _a set of non-branching geodesics_ if
and only if for any $\gamma^{1},\gamma^{2}\in\Gamma$, it holds:

$\exists\;\bar{t}\in(0,1)\text{ such that
}\;\forall\,t\in[0,\bar{t}\,]\quad\gamma_{t}^{1}=\gamma_{t}^{2}\quad\Longrightarrow\quad\gamma^{1}_{s}=\gamma^{2}_{s},\quad\forall\,s\in[0,1].$
(3.6)

It is clear that if $(X,\mathsf{d})$ is a smooth Riemannian manifold, then any
subset $\Gamma\subset\mathrm{Geo}(X)$ is a set of non branching geodesics.
This is true, more generally, if $(X,\mathsf{d})$ is an ideal sub-Riemannian
manifold, i.e. admitting no non-trivial abnormal geodesics: in fact, a sub-
Riemannian geodesic can branch only if it contains an abnormal segment, cf.
[67].

We endow the metric space $(X,\mathsf{d})$ with a measure
$\mathfrak{m}\in\mathscr{M}_{\mathrm{loc}}(X)$, i.e. non-negative and finite
on bounded sets. The triple $(X,\mathsf{d},\mathfrak{m})$ is called _metric
measure space_ (m.m.s. for short). We denote with
$\mathcal{P}_{ac}(X,\mathfrak{m})\subset\mathcal{P}(X)$ the subspace of
probability measures which are absolutely continuous with respect to the
reference measure $\mathfrak{m}$.

A metric measure space $(X,\mathsf{d},\mathfrak{m})$ is _essentially non-
branching_ if and only if for any
$\mu_{0},\mu_{1}\in\mathcal{P}_{2}(X,\mathsf{d})\cap\mathcal{P}_{ac}(X,\mathfrak{m})$,
any $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ is concentrated on a set
$\Gamma\subset\mathrm{Geo}(X)$ of non-branching geodesics.

####  3.1.3 Relative entropies

For $\mu\in\mathcal{P}_{2}(X)$, we define its relative (Boltzmann-Shannon)
entropy by

$\mathrm{Ent}(\mu|\mathfrak{m}):=\int_{X}\rho\log(\rho)\,\mathfrak{m},\qquad\text{
if
$\mu=\rho\,\mathfrak{m}\in\mathcal{P}_{2}(X)\cap\mathcal{P}_{ac}(X,\mathfrak{m})$},$
(3.7)

in case $\rho\log(\rho)\in L^{1}(X,\mathfrak{m})$, otherwise we set
$\mathrm{Ent}(\mu|\mathfrak{m}):=+\infty$.

Since $(0,+\infty)\ni x\mapsto x\log x$ is convex, Jensen’s inequality gives

$\mathrm{Ent}(\mu|\mathfrak{m})\geq-\log\mathfrak{m}(\operatorname{\mathrm{supp}}\mu),\qquad\forall\,\mu\in\mathcal{P}(X)\text{
with }\mathfrak{m}(\operatorname{\mathrm{supp}}\mu)<\infty.$ (3.8)

We set
$\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m})):=\\{\mu\in\mathcal{P}_{2}(X)\mid\mathrm{Ent}(\mu|\mathfrak{m})\in\mathbb{R}\\}$
to be the finiteness domain of the entropy and

$\displaystyle\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$
$\displaystyle:=\\{\mu\in\mathcal{P}_{bs}(X,\mathsf{d})\mid\operatorname{\mathrm{supp}}\,\mu\subseteq\operatorname{\mathrm{supp}}\,\mathfrak{m}\\},$
$\displaystyle\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$
$\displaystyle:=\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\cap\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m}).$

In order to formulate “dimensional” Ricci curvature lower bounds, it is
convenient to introduce also the following dimensional entropy (cf. [44]):

$\mathrm{U}_{n}(\mu|\mathfrak{m}):=\exp\left(-\frac{\mathrm{Ent}(\mu|\mathfrak{m})}{n}\right),\qquad
n\in[1,+\infty),$ (3.9)

with the understanding that $\mathrm{U}_{n}(\mu|\mathfrak{m}):=0$ if
$\mu\notin\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))$.

###  3.2 Gauge functions on metric spaces

In order to obtain a unified framework

  * •

embracing both sub-Riemannian structures and Lott-Sturm-Villani’s
$\mathsf{CD}$ m.m.s.,

  * •

yielding sharp geometric and functional inequalities,

we add an additional structure to a metric measure space
$(X,\mathsf{d},\mathfrak{m})$, that is a non-negative Borel function, called
_gauge function_ :

$\mathsf{G}:X\times X\to[0,+\infty].$ (3.10)

The following analogy explains the role of the gauge function. A Riemannian
manifold $(M,g)$ has Ricci curvature bounded from below if there exists
$K\in\mathbb{R}$ such that, for all $x,y\in M$ and any geodesic $\gamma$
between $x$ and $y$ it holds

$\mathrm{Ric}(\dot{\gamma},\dot{\gamma})\geq
K\|\dot{\gamma}\|^{2}=K\,\mathsf{d}(\gamma_{0},\gamma_{1})^{2}.$ (3.11)

The distance function $\mathsf{d}$ is used in the right hand side as a _gauge_
to measure the extent of the lower Ricci curvature bound, quantified by the
constant $K$. The idea is to replace the distance $\mathsf{d}$ with a general
gauge function $\mathsf{G}$ for curvature bounds.

Gauge functions will be key in our extension of the synthetic theory of
curvature bounds to the sub-Riemannian setting, where it is well-known that
the effect of the curvature on transport inequalities is not expressed via the
distance but rather via other intrinsically sub-Riemannian functions (a
phenomenon observed in [19], [24, Sec. 8.1]).

###  3.3 Curvature-dimension conditions for gauge spaces

Let $\mathsf{s}:[0,+\infty)\to\mathbb{R}$ be a continuous function, with
$N\in[1,+\infty)$ such that

$\mathsf{s}(\theta)=c\,\theta^{N}+o(\theta^{N})\qquad\text{ for some }c>0.$
(3.12)

The parameter $N$ will be the sharp upper bound for a new notion of dimension,
which is in general different from the Hausdorff one, see Section 4.5. Denote:

$\mathcal{D}:=\inf\\{\theta>0\mid\mathsf{s}(\theta)=0\\}.$ (3.13)

From the assumptions on $\mathsf{s}$ it is clear that $\mathcal{D}>0$. The
parameter $\mathcal{D}$ will give a sharp upper bound on the gauge function,
see Section 4.3. Define the _distortion coefficient_
$\beta_{(\cdot)}(\cdot):[0,1]\times[0,+\infty]\to[0,+\infty]$ as

$\displaystyle(t,\theta)\in[0,1]\times[0,+\infty]\mapsto\beta_{t}(\theta):=\begin{cases}t^{N}&\theta=0,\\\
\dfrac{\mathsf{s}(t\theta)}{\mathsf{s}(\theta)}&0<\theta<\mathcal{D},\\\
\displaystyle\liminf_{\phi\to\mathcal{D}^{-}}\dfrac{\mathsf{s}(t\phi)}{\mathsf{s}(\phi)}&\theta\geq\mathcal{D}.\end{cases}$
(3.14)

######  Remark 3.3.

In applications to sub-Riemannian geometry, the function $\mathsf{s}$ is
chosen in a class of models which are characterized as solutions to suitable
ODEs, see Proposition 8.5 and Remark 8.6. However, here we develop the theory
in full generality.

We collect some elementary properties following from the definition.

######  Proposition 3.4.

Any distortion coefficient satisfies the following:

  1. (i)

$\beta_{0}(\theta)=0$ and $\beta_{1}(\theta)=1$ for all
$\theta\in[0,+\infty]$;

  2. (ii)

if $\mathcal{D}<+\infty$, then $\beta_{t}(\theta)=+\infty$ for all
$\theta\geq\mathcal{D}$ and $t\in(0,1)$;

  3. (iii)

$\beta_{t}(\theta)=0$ for some $\theta<+\infty$ if and only if $t=0$;

  4. (iv)

for every $t\in[0,1]$, $\theta_{j},\theta\in[0,+\infty)$, if
$\theta_{j}\to\theta$ then $\beta_{t}(\theta_{j})\to\beta_{t}(\theta)$ in
$[0,+\infty]$;

  5. (v)

$\beta$ is continuous and finite when restricted to
$[0,1]\times[0,\mathcal{D})$;

  6. (vi)

for all fixed $t\in(0,1]$, the distortion coefficient $\beta_{t}(\cdot)$ is
bounded from below away from zero on any bounded set of $[0,+\infty)$;

  7. (vii)

assume that $\mathcal{D}<+\infty$, and that $\mathsf{s}\in
C^{\omega}([0,\mathcal{D}])$. Then there exists $N^{\prime}\geq N$ such that
$\beta_{t}(\theta)\geq t^{N^{\prime}}$ for all $t\in[0,1]$ and
$\theta\in[0,+\infty]$.

######  Definition 3.5.

Let $n\in[1,+\infty)$, and $\beta$ as in (3.14). We say that a metric measure
space $(X,\mathsf{d},\mathfrak{m})$ with gauge $\mathsf{G}$ satisfies:

  * •

$\mathsf{CD}(\beta,n)$ if for all
$\mu_{0}\in\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$,
$\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$,
there exists a $W_{2}$-geodesic
$(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(X,\mathsf{d})$ connecting them,
induced by $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$, such that it holds

$\mathrm{U}_{n}(\mu_{t}|\mathfrak{m})\geq\exp\left(\frac{1}{n}\int_{\mathrm{Geo}(X)}\log\beta_{1-t}\big{(}\mathsf{G}(\gamma_{1},\gamma_{0}))\,\nu(\mathrm{d}\gamma)\right)\mathrm{U}_{n}(\mu_{0}|\mathfrak{m})\\\
+\exp\left(\frac{1}{n}\int_{\mathrm{Geo}(X)}\log\beta_{t}\big{(}\mathsf{G}(\gamma_{0},\gamma_{1})\big{)}\,\nu(\mathrm{d}\gamma)\right)\mathrm{U}_{n}(\mu_{1}|\mathfrak{m}),\qquad\forall\,t\in(0,1),$
(3.15)

with the convention that $\infty\cdot 0=0$. We say that $\mathsf{CD}(\beta,n)$
is satisfied _in the strong sense_ if (3.15) is satisfied for all
$W_{2}$-geodesics connecting $\mu_{0}$ and $\mu_{1}$.

  * •

$\mathsf{MCP}(\beta)$ if for any
$\bar{x}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ and
$\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\bar{x}\notin\operatorname{\mathrm{supp}}\mu_{1}$ there exists a
$W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(X,\mathsf{d})$
from $\mu_{0}=\delta_{\bar{x}}$ to $\mu_{1}$ such that

$\mathrm{U}_{n}(\mu_{t}|\mathfrak{m})\geq\exp\left(\frac{1}{n}\int_{X}\log\beta_{t}\big{(}\mathsf{G}(\bar{x},x)\big{)}\,\mu_{1}(\mathrm{d}x)\right)\mathrm{U}_{n}(\mu_{1}|\mathfrak{m}),\qquad\forall\,t\in(0,1),$
(3.16)

for some (and then every) $n\geq 1$.

######  Remark 3.6.

Notice that (3.16) is equivalent to

$\mathrm{Ent}(\mu_{t}|\mathfrak{m})\leq\mathrm{Ent}(\mu_{1}|\mathfrak{m})-\int_{X}\log\beta_{t}\big{(}\mathsf{G}(\bar{x},x)\big{)}\,\mu_{1}(\mathrm{d}x),\qquad\forall\,t\in(0,1),$
(3.17)

so that (3.16) does not depend on the value of $n$.

######  Remark 3.7 ($\mathsf{CD}$ implies $\mathsf{MCP}$).

It is clear that, with our definitions, $\mathsf{CD}(\beta,n)$ for some
$n\in[1,+\infty)$ implies $\mathsf{MCP}(\beta)$, since one can choose in
(3.15) $\mu_{0}$ equal to a Dirac mass.

######  Remark 3.8 (About
$\mu_{0}\in\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$).

In case the gauge function $\mathsf{G}:X\times X\to[0,+\infty)$ is
_continuous_ , via a standard approximation argument it is possible to see
that the $\mathsf{CD}(\beta,n)$ condition as stated in Definition 3.5 is
equivalent to the following: for all
$\mu_{0},\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$,
there exists a $W_{2}$-geodesic
$(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(X,\mathsf{d})$ connecting them,
induced by $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$, such that (3.15) holds. The
latter formulation is closer in spirit to Lott-Sturm-Villani’s curvature-
dimension conditions for metric measure spaces; note that in this case
$\mathsf{G}=\mathsf{d}$ is of course continuous. When $\mathsf{G}$ is not
continuous, the flexibility to take
$\mu_{0}\in\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$ turns out to be
technically convenient; for instance it makes neat the implication
$\mathsf{CD}(\beta,n)\Rightarrow\mathsf{MCP}(\beta)$, implication which would
not be clear otherwise.

######  Remark 3.9 (Length space properties for
$\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\subset\mathcal{P}_{2}(X,\mathsf{d})$
and $\operatorname{\mathrm{supp}}\mathfrak{m}$).

The fact that $(X,\mathsf{d},\mathfrak{m})$ satisfies $\mathsf{CD}(\beta,n)$
implies that
$\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\subset\mathcal{P}_{2}(X,\mathsf{d})$
is a length space (with metric $W_{2}$) and that
$\operatorname{\mathrm{supp}}\mathfrak{m}$ is a length space (with metric
$\mathsf{d}$). The latter statement holds also if
$(X,\mathsf{d},\mathfrak{m})$ satisfies $\mathsf{MCP}(\beta)$. The proof of
these statements follow verbatim [81, Remark 4.6, (iii)] and thus is omitted.
We stress that these facts do not use the inequalities (3.15) or (3.16), but
only the existence of $W_{2}$-geodesics between suitable pairs of measures.
Moreover, as a consequence of [36, Thm. 1.1] it follows that: if
$(X,\mathsf{d},\mathfrak{m})$ is an essentially non-branching
$\mathsf{MCP}(K,N)$ space, then
$\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\subset\mathcal{P}_{2}(X,\mathsf{d})$
is a length space (with metric $W_{2}$). Along the same lines, one can prove
that
$\mathrm{Dom}(\mathrm{Ent}(\cdot|\mathfrak{m}))\subset\mathcal{P}_{2}(X,\mathsf{d})$
is a length space under the assumption that $(X,\mathsf{d},\mathfrak{m})$ is
an essentially non-branching $\mathsf{MCP}(\beta)$ space.

Fix a metric measure space $(X,\mathsf{d},\mathfrak{m})$ with gauge function
$\mathsf{G}$. The next proposition contains interpolation and monotonicity
properties of the $\mathsf{CD}$ classes.

######  Proposition 3.10.

Let $\alpha,\beta,\beta_{0},\beta_{1}$ distortion coefficients as in (3.14),
let $n,n_{0},n_{1}\geq 1$, and $\theta\in[0,1]$. Then the following properties
hold

  1. (i)

$\mathrm{CD}(\beta_{0},n_{0})\cap\mathrm{CD}(\beta_{1},n_{1})$ with at least
one of the two being satisfied in the strong sense $\Longrightarrow$
$\mathrm{CD}(\beta_{\theta},n_{\theta})$ where we have set

$\beta_{\theta}:=\beta_{0}^{1-\theta}\beta_{1}^{\theta},\qquad
n_{\theta}:=(1-\theta)n_{0}+\theta n_{1};$

  2. (ii)

if $m\geq 0$. Then $\mathrm{CD}(\beta,n)$ $\Longrightarrow$
$\mathrm{CD}(t^{m}\beta,n+m)$;

  3. (iii)

if $\beta\geq\alpha$, then $\mathrm{CD}(\beta,n)$ $\Longrightarrow$
$\mathrm{CD}(\alpha,n)$.

###### Proof.

Proof of (i). Observe that
$\mathrm{\mathrm{U}}_{n}=(\mathrm{U}_{m})^{\frac{m}{n}}$ for all $n,m\geq 1$.
Hence

$\mathrm{U}_{n_{\theta}}=\mathrm{U}_{n_{\theta}}^{(1-\theta)}\mathrm{U}_{n_{\theta}}^{\theta}=\mathrm{U}_{n_{0}}^{(1-\theta)\frac{n_{0}}{n_{\theta}}}\mathrm{U}_{n_{1}}^{\theta\frac{n_{1}}{n_{\theta}}}.$
(3.18)

Since at least one of the two $\mathsf{CD}(\beta_{i},n_{i})$ conditions is
satisfied in the strong sense, we may assume that that for any given
$\mu_{0}\in\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$,
$\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$,
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$
there exists a $W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ such that

$\mathrm{U}_{n_{i}}(\mu_{t}|\mathfrak{m})\geq\exp\left(\frac{1}{n_{i}}\int_{\mathrm{Geo}(X)}\log\beta_{i,1-t}\big{(}\mathsf{G}(\gamma_{1},\gamma_{0})\big{)}\,\nu(\mathrm{d}\gamma)\right)\mathrm{U}_{n_{i}}(\mu_{0}|\mathfrak{m})\\\
+\exp\left(\frac{1}{n_{i}}\int_{\mathrm{Geo}(X)}\log\beta_{i,t}\big{(}\mathsf{G}(\gamma_{0},\gamma_{1})\big{)}\,\nu(\mathrm{d}\gamma)\right)\mathrm{U}_{n_{i}}(\mu_{1}|\mathfrak{m}),\qquad\forall\,t\in(0,1),\;i=0,1.$
(3.19)

For $i=0,1$ and $t\in(0,1)$, set

$\displaystyle a_{i,t}$
$\displaystyle:=\exp\left(\frac{1}{n_{i}}\int_{\mathrm{Geo}(X)}\log\beta_{i,1-t}\big{(}\mathsf{G}(\gamma_{1},\gamma_{0})\big{)}\,\nu(\mathrm{d}\gamma)\right),$
$\displaystyle b_{i,t}$
$\displaystyle:=\exp\left(\frac{1}{n_{i}}\int_{\mathrm{Geo}(X)}\log\beta_{i,t}\big{(}\mathsf{G}(\gamma_{0},\gamma_{1})\big{)}\,\nu(\mathrm{d}\gamma)\right).$

Omitting $\mathfrak{m}$ from the notation, we have

$\displaystyle\mathrm{U}_{n_{\theta}}(\mu_{t})$
$\displaystyle=\mathrm{U}_{n_{0}}(\mu_{t})^{(1-\theta)\tfrac{n_{0}}{n_{\theta}}}\mathrm{U}_{n_{1}}(\mu_{t})^{\theta\tfrac{n_{1}}{n_{\theta}}}$
(3.20)
$\displaystyle\geq\left[a_{0,t}\mathrm{U}_{n_{0}}(\mu_{0})+b_{0,t}\mathrm{U}_{n_{0}}(\mu_{1})\right]^{(1-\theta)\tfrac{n_{0}}{n_{\theta}}}\left[a_{1,t}\mathrm{U}_{n_{1}}(\mu_{0})+b_{1,t}\mathrm{U}_{n_{1}}(\mu_{1})\right]^{\theta\tfrac{n_{1}}{n_{\theta}}}.$
(3.21)

Now we use the Hölder inequality: for $\frac{1}{p}+\frac{1}{q}=1$ we have for
positive $x_{j},y_{j}$

$\left(\sum_{j=1}^{m}x_{j}\right)^{1/p}\left(\sum_{j=1}^{m}y_{j}\right)^{1/q}\geq\sum_{j=1}^{m}x_{j}^{1/p}y_{j}^{1/q}.$
(3.22)

Hence we obtain

$\mathrm{U}_{n_{\theta}}(\mu_{t})\geq
a_{0,t}^{(1-\theta)\tfrac{n_{0}}{n_{\theta}}}a_{1,t}^{\theta\tfrac{n_{1}}{n_{\theta}}}\mathrm{U}_{n_{\theta}}(\mu_{0})+b_{0,t}^{(1-\theta)\tfrac{n_{0}}{n_{\theta}}}b_{1,t}^{\theta\tfrac{n_{1}}{n_{\theta}}}\mathrm{U}_{n_{\theta}}(\mu_{1}).$
(3.23)

Which corresponds to the desired inequality for the condition
$\mathsf{CD}(\beta_{\theta},n_{\theta})$.

Proof of (ii). We observe first that the map
$\mathbb{R}^{2}\ni(x,y)\mapsto\log(e^{x}+e^{y})$ is convex, and thus for
$n\geq 1$, $m\geq 0$, all $t\in(0,1)$ and $x,y\in\mathbb{R}$ it holds

$\displaystyle\frac{n}{n+m}\log(e^{x}+e^{y})$
$\displaystyle=\frac{n}{n+m}\log(e^{x}+e^{y})+\frac{m}{n+m}\log\left(e^{\log(1-t)}+e^{\log
t}\right)$ (3.24)
$\displaystyle\geq\log\left(e^{\tfrac{n}{n+m}x+\tfrac{m}{m+n}\log(1-t)}+e^{\tfrac{n}{n+m}y+\tfrac{m}{n+m}\log
t}\right).$ (3.25)

Fix a $W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ for which the
$\mathsf{CD}(\beta,n)$ inequality holds true. Observe that:

$\displaystyle\log\mathrm{U}_{n+m}(\mu_{t})$
$\displaystyle=\frac{n}{n+m}\log\mathrm{U}_{n}(\mu_{t}),\qquad\forall\,t\in[0,1].$
(3.26)

Using the $\mathsf{CD}(\beta,n)$ inequality in the previous identity and
applying (3.25), we immediately obtain the validity of the
$\mathsf{CD}(t^{m}\beta,n+m)$ condition.

Proof of (iii). Obvious. ∎

######  Proposition 3.11.

Let $(X,\mathsf{d},\mathfrak{m})$ be essentially non-branching. Assume that
$(X,\mathsf{d},\mathfrak{m})$ with gauge $\mathsf{G}=\mathsf{d}$ satisfies the
$\mathsf{MCP}(\beta)$, for $\beta$ defined as in (3.14) in terms of a function
$\mathsf{s}$ as in (3.12), with order $N$ at zero. If $\mathcal{D}$ is finite
and $\mathsf{s}\in C^{\omega}([0,\mathcal{D}])$, then there exists
$N^{\prime}\geq N$ such that $(X,\mathsf{d},\mathfrak{m})$ satisfies the
$\mathsf{MCP}(t^{N^{\prime}})$ with gauge $\mathsf{G}=\mathsf{d}$ and thus
$(X,\mathsf{d},\mathfrak{m})$ satisfies the classical measure contraction
property $\mathsf{MCP}(0,N^{\prime})$.

###### Proof.

By Proposition 3.4(vii), there exists $N^{\prime}\geq N$ such that
$\beta_{t}\geq t^{N^{\prime}}$ for all $t\in[0,1]$. Hence
$\mathsf{MCP}(t^{N^{\prime}})$ holds and we conclude by Theorem 3.12(ii). ∎

####  3.3.1 Recovering the Lott-Sturm-Villani’s theory

In this section we explain how our general theory of curvature bounds for
gauge spaces contains the classical curvature-dimension theory for m.m.s. à la
Lott-Sturm-Villani. In this section we consider $K\in\mathbb{R}$ and
$N\in[1,+\infty)$. Set

$[0,+\infty]\ni\theta\mapsto\mathsf{s}_{K,N}(\theta)=\begin{cases}\sin\left(\theta\sqrt{K/N)}\right)^{N}&K>0,\\\
\theta^{N}&K=0,\\\ \sinh\left(\theta\sqrt{-K/N}\right)^{N}&K<0.\end{cases}$
(3.27)

Notice that $\mathsf{s}_{K,N}(\theta)$ has order $N$ as $\theta\to 0$ for all
$K\in\mathbb{R}$. Moreover,

$\mathcal{D}_{K,N}=\inf\\{\theta>0\mid\mathsf{s}_{K,N}(\theta)=0\\}=\begin{cases}\frac{\pi}{\sqrt{K/N}}&K>0,\\\
+\infty&K\leq 0.\end{cases}$ (3.28)

The corresponding distortion coefficient, following the recipe of (3.14), is
then

$[0,1]\times[0,+\infty]\ni(t,\theta)\mapsto(\beta_{K,N}^{\sigma})_{t}(\theta)=\begin{cases}+\infty&K\theta^{2}\geq
N\pi^{2},\\\
\frac{\sin\left(t\theta\sqrt{K/N}\right)^{N}}{\sin\left(\theta\sqrt{K/N}\right)^{N}}&0<K\theta^{2}<N\pi^{2},\\\
t^{N}&K\theta^{2}=0,\\\
\frac{\sinh\left(t\theta\sqrt{|K|/N}\right)^{N}}{\sinh\left(\theta\sqrt{|K|/N}\right)^{N}}&K\theta^{2}<0.\end{cases}$
(3.29)

The notation is motivated by the fact that

$(\beta_{K,N}^{\sigma})_{t}(\theta)=\left(\sigma_{K,N}^{(t)}(\theta)\right)^{N},\qquad\forall\,t\in[0,1],\quad\forall\,\theta\in[0,+\infty),$
(3.30)

where $\sigma_{K,N}^{(\cdot)}(\cdot)$ are the functions considered in [17, 44]
(note that [44] used the notation $\sigma_{K/N}^{(\cdot)}(\cdot)$). Consider
also

$[0,1]\times[0,+\infty]\ni(t,\theta)\mapsto(\beta_{K,N}^{\tau})_{t}(\theta)=\begin{cases}+\infty&K\theta^{2}\geq(N-1)\pi^{2},\\\
t\frac{\sin\left(t\theta\sqrt{K/(N-1)}\right)^{N-1}}{\sin\left(\theta\sqrt{K/(N-1)}\right)^{N-1}}&0<K\theta^{2}<(N-1)\pi^{2},\\\
t^{N}&K\theta^{2}=0,\\\
t\frac{\sinh\left(t\theta\sqrt{|K|/(N-1)}\right)^{N-1}}{\sinh\left(\theta\sqrt{|K|/(N-1)}\right)^{N-1}}&K\theta^{2}<0.\end{cases}$
(3.31)

Again, the notation is motivated by the fact that

$(\beta_{K,N}^{\tau})_{t}(\theta)=\big{(}\tau_{K,N}^{(t)}(\theta)\big{)}^{N},\qquad\forall\,t\in[0,1],\quad\forall\,\theta\in[0,+\infty),$
(3.32)

where $\tau_{K,N}^{(t)}(\theta):=t^{1/N}\sigma^{(t)}_{K,N-1}(\theta)^{1-1/N}$
are the functions considered in [82].

######  Theorem 3.12.

Let $(X,\mathsf{d},\mathfrak{m})$ be an essentially non-branching m.m.s.
equipped with the gauge function $\mathsf{G}=\mathsf{d}$. The following hold:

  1. (i)

$(X,\mathsf{d},\mathfrak{m})$ satisfies the entropic curvature-dimension
condition $\mathsf{CD}^{e}(K,N)$ of Erbar-Kuwada-Sturm [44] if and only if
$(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies
$\mathsf{CD}(\beta_{K,N}^{\sigma},N)$ in the sense of Definition 3.5;

  2. (ii)

$(X,\mathsf{d},\mathfrak{m})$ satisfies the measure contraction property
$\mathsf{MCP}(K,N)$ in the formulation of Ohta [71] if and only if
$(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies
$\mathsf{MCP}(\beta_{K,N}^{\tau})$ in the sense of Definition 3.5.

###### Proof.

Proof of (i),
$\mathsf{CD}(\beta_{K,N}^{\sigma},N)\,\Rightarrow\,\mathsf{CD}^{e}(K,N)$. The
map $\theta\mapsto\log\left[(\beta_{K,N}^{\sigma})_{t}(\sqrt{\theta})\right]$
is convex. Using Jensen’s inequality, it follows that

$\displaystyle\frac{1}{N}\int_{\mathrm{Geo}(X)}\log(\beta_{K,N}^{\sigma})_{t}(\mathsf{d}(\gamma_{1},\gamma_{0}))\nu(\mathrm{d}\gamma)$
$\displaystyle\geq\frac{1}{N}\log(\beta_{K,N}^{\sigma})_{t}\left(\sqrt{\int_{\mathrm{Geo}(X)}\mathsf{d}(\gamma_{1},\gamma_{0})^{2}\nu(\mathrm{d}\gamma)}\right)$
(3.33)
$\displaystyle=\frac{1}{N}\log\left[(\beta_{K,N}^{\sigma})_{t}(W_{2}(\mu_{0},\mu_{1}))\right]$
(3.34) $\displaystyle=\log\sigma_{K/N}^{(t)}(W_{2}(\mu_{0},\mu_{1}))\,.$
(3.35)

Thus, if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space satisfying
the $\mathsf{CD}(\beta,N)$ condition for the Gauge function
$\mathsf{G}=\mathsf{d}$, then the entropic $\mathsf{CD}^{e}(K,N)$ convexity
inequality of [44] is satisfied for measures
$\mu_{0},\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$.
A straightforward approximation argument, yields the validity of the
$\mathsf{CD}^{e}(K,N)$ convexity inequality for measures
$\mu_{0},\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\mathfrak{m}(\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1})=0$.
At this point, one can follow verbatim the proof of [44, Thm. 3.12, (iii)
$\Rightarrow$ (ii)] (noticing that there the $\mathsf{CD}^{e}(K,N)$ convexity
inequality is used only for measures whose supports have
$\mathfrak{m}$-negligible intersection) and infer that for any
$\mu_{0},\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ (without
any condition on the intersection of the supports) there exists $\nu\in{\rm
OptGeo}(\mu_{0},\mu_{1})$ such that $({\rm e}_{t})_{\sharp}\nu\ll\mathfrak{m}$
for all $t\in(0,1)$ and

$\rho_{t}(\gamma_{t})^{-\frac{1}{N}}\geq\sigma_{K/N}^{(1-t)}(\mathsf{d}(\gamma_{0},\gamma_{1}))\rho_{0}(\gamma_{0})^{-\frac{1}{N}}+\sigma_{K/N}^{(t)}(\mathsf{d}(\gamma_{0},\gamma_{1}))\rho_{1}(\gamma_{1})^{-\frac{1}{N}},$
(3.36)

for $\nu$-a.e. $\gamma\in\mathrm{Geo}(X)$, where $\rho_{t}$ is the density of
$({\rm e}_{t})_{\sharp}\nu$ w.r.t. $\mathfrak{m}$. Using now [44, Thm. 3.12,
(ii) $\Rightarrow$ (iii)] we infer that $(X,\mathsf{d},\mathfrak{m})$
satisfies the entropic curvature-dimension condition $\mathsf{CD}^{e}(K,N)$.

Proof of (i),
$\mathsf{CD}^{e}(K,N)\,\Rightarrow\,\mathsf{CD}(\beta_{K,N}^{\sigma},N)$.
Conversely, if $(X,\mathsf{d},\mathfrak{m})$ is essentially non-branching and
it satisfies $\mathsf{CD}^{e}(K,N)$, then from [44, Thm. 3.12, (iii)
$\Rightarrow$ (ii)] we know that for any
$\mu_{0},\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ there
exists $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ such that $({\rm
e}_{t})_{\sharp}\nu\ll\mathfrak{m}$ for all $t\in(0,1)$ and (3.36) hold for
$\nu$-a.e. $\gamma\in\mathrm{Geo}(X)$, where $\rho_{t}$ is the density of
$({\rm e}_{t})_{\sharp}\nu$ w.r.t. $\mathfrak{m}$. With the same argument one
can prove that (3.36) holds more generally when
$\mu_{0}\in\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$, up to removing the
first term in the right hand side (we will assume this convention throughout
the proof). We rewrite (3.36) as

$-\frac{1}{N}\log\rho_{t}(\gamma_{t})\geq\log\left[\exp\left(-\frac{1}{N}\log\rho_{0}(\gamma_{0})+\log
a_{1-t}(\gamma_{0},\gamma_{1})\right)\right.\\\
\left.+\exp\left(-\frac{1}{N}\log\rho_{1}(\gamma_{1})+\log
a_{t}(\gamma_{0},\gamma_{1})\right)\right],$ (3.37)

where we set
$a_{t}(\gamma_{0},\gamma_{1})=\sigma_{K/N}^{(t)}(\mathsf{d}(\gamma_{0},\gamma_{1}))$.
Recalling that the map $(x,y)\mapsto\log(e^{x}+e^{y})$ is convex, integrating
(3.37) w.r.t. $\nu$ and using Jensen’s inequality, we obtain

$-\frac{1}{N}\mathrm{Ent}(\mu_{t}|\mathfrak{m})\geq\log\left[\exp\left(-\frac{1}{N}\mathrm{Ent}(\mu_{0}|\mathfrak{m})+\int_{\mathrm{Geo}(X)}\log
a_{1-t}(\gamma_{0},\gamma_{1})\nu(\mathrm{d}\gamma)\right)\right.\\\
\left.+\left(-\frac{1}{N}\mathrm{Ent}(\mu_{1}|\mathfrak{m})+\int_{\mathrm{Geo}(X)}\log
a_{t}(\gamma_{0},\gamma_{1})\nu(\mathrm{d}\gamma)\right)\right].$ (3.38)

Taking the exponential of (3.38) and recalling (3.30), we obtain that
$(X,\mathsf{d},\mathfrak{m})$ endowed with the gauge function
$\mathsf{G}=\mathsf{d}$ satisfies the $\mathsf{CD}(\beta_{K,N}^{\sigma},N)$
condition (3.15).

Proof of (ii), $\mathsf{MCP}(\beta_{K,N}^{\tau})\Rightarrow\mathsf{MCP}(K,N)$.
Using (3.32) and arguing along the lines of the proof of [44, Thm. 3.12, (iii)
$\Rightarrow$ (ii)], one uses the essential non-branching assumption to
localize the $\mathsf{MCP}(\beta_{K,N}^{\tau})$ integral estimate to a
pointwise estimate along geodesics. More precisely, one can show that for
every $\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ and every
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}\setminus\operatorname{\mathrm{supp}}\mu_{1}$
there exists $\nu\in{\rm OptGeo}(\delta_{x_{0}},\mu_{1})$ such that $({\rm
e}_{t})_{\sharp}\nu\ll\mathfrak{m}$ for all $t\in(0,1)$ and

$\rho_{t}(\gamma_{t})\leq\tau_{K,N}^{(t)}(\mathsf{d}(\gamma_{0},\gamma_{1}))^{-N}\rho_{1}(\gamma_{1}),\qquad\nu\text{-a.e.
}\gamma\in\mathrm{Geo}(X),$ (3.39)

where $\rho_{t}$ denotes the density of $({\rm e}_{t})_{\sharp}\nu$ w.r.t.
$\mathfrak{m}$. Let $A$ be a bounded Borel set with positive measure and let
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}\setminus\bar{A}$, where
$\bar{A}$ is the closure of $A$. Letting
$\mu_{1}:=\mathfrak{m}(A)^{-1}\,\mathfrak{m}\llcorner_{A}$ and applying
(3.39), one obtains Ohta’s [71] formulation of $\mathsf{MCP}(K,N)$

$\mathfrak{m}\llcorner_{A}\geq({\rm
e}_{t})_{\sharp}\left(\tau_{K,N}^{(t)}\big{(}\mathsf{d}(\gamma_{0},\gamma_{1})\big{)}^{N}\mathfrak{m}(A)\nu\right),$
(3.40)

with the caveat
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}\setminus\bar{A}$ and $A$
bounded. The general case follows by a standard approximation argument, that
we sketch below.

Observing that the distance is a _meek_ gauge function in the sense of
Definition 4.22, the generalized Bishop-Gromov Theorem 4.24 holds. We infer
that $\mathfrak{m}(\\{x\\})=0$ for every $x\in X$ and that
$(\operatorname{\mathrm{supp}}\mathfrak{m},\mathsf{d})$ is a proper metric
space (see Corollary 4.15).

Let $A$ be a bounded Borel set with positive measure and let
$x_{0}\in\bar{A}\,\cap\,\operatorname{\mathrm{supp}}\mathfrak{m}$. For
$\varepsilon>0$ small enough we have that $\mathfrak{m}(A\setminus
B_{\varepsilon}(x_{0}))>0$. Set

$\mu_{1,\varepsilon}:=(\mathfrak{m}(A\setminus
B_{\varepsilon}(x_{0})))^{-1}\,\mathfrak{m}\llcorner_{A\setminus
B_{\varepsilon}(x_{0})}.$ (3.41)

From the discussion above, we know that there exists $\nu_{\varepsilon}\in{\rm
OptGeo}(\delta_{x_{0}},\mu_{1,\varepsilon})$ such that

$\mathfrak{m}\llcorner_{A\setminus B_{\varepsilon}(x_{0})}\geq({\rm
e}_{t})_{\sharp}\left(\tau_{K,N}^{(t)}\big{(}\mathsf{d}(\gamma_{0},\gamma_{1})\big{)}^{N}\mathfrak{m}\big{(}A\setminus
B_{\varepsilon}(x_{0})\big{)}\nu_{\varepsilon}\right).$ (3.42)

By letting $\varepsilon\downarrow 0$, one obtains that there exists
$\nu\in{\rm OptGeo}(\delta_{x_{0}},\mu_{1})$ such that (3.40) holds. For a
general Borel set $A$ of finite positive measure: intersect $A$ with the
metric ball $B_{R}(x_{0})$, obtain (3.42) and take the limit as
$R\uparrow+\infty$ to conclude.

Proof of (ii), $\mathsf{MCP}(K,N)\Rightarrow\mathsf{MCP}(\beta_{K,N}^{\tau})$.
Conversely, it is standard to check that (3.40) implies (3.39) (see for
instance [36, App. A]). Arguing along the lines of
$\mathsf{CD}^{e}(K,N)\,\Rightarrow\,\mathsf{CD}(\beta_{K,N}^{\sigma},N)$, we
obtain that $(X,\mathsf{d},\mathfrak{m})$ endowed with the gauge function
$\mathsf{G}=\mathsf{d}$ satisfies the $\mathsf{MCP}(\beta_{K,N}^{\tau})$
condition (3.16). ∎

######  Remark 3.13 (Connection with other synthetic formulations).

Let $K\in\mathbb{R},N\in[1,+\infty)$. In case $(X,\mathsf{d},\mathfrak{m})$ is
essentially non-branching:

  * •

It was proved in [44, Thm. 3.12] that the reduced $\mathsf{CD}^{*}(K,N)$
condition (see [17]) is equivalent to the entropic $\mathsf{CD}^{e}(K,N)$.

  * •

The $\mathsf{CD}(K,N)$ condition (as in [82]) is equivalent to
$\mathsf{CD}^{*}(K,N)$. This is a deep result establishing the local-to-global
property for $\mathsf{CD}(K,N)$; it was proved in [35] in case
$\mathfrak{m}(X)<\infty$. Their approach is based on the _needle
decomposition_ of [57] and the _localization technique_ developed in [37].

  * •

Ohta’s [71] and Sturm’s [82] formulations of $\mathsf{MCP}(K,N)$ are
equivalent, see for instance [36, App. A].

  * •

Our theory is modeled on the convexity inequalities for the entropic
curvature-dimension condition à la Erbar-Kuwada-Sturm [44], that involves the
functional $\mathrm{U}_{N}$. One could have developed the theory studying
instead the Renyi entropy functional, as in [82, 17], or general functionals
in the McCann class $\mathcal{DC}(N)$ as in [64, 63], with analogous outcomes.
For a discussion of connections between the approach in [64, 63] and the
$\mathsf{CD}^{*}(K,N)$ condition see for instance [14, Sec. 9].

######  Remark 3.14 (Connection with E. Milman’s theory).

In [68], E. Milman introduced a quasi-convex relaxation of the
Lott–Sturm–Villani’s $\mathsf{CD}(K,N)$, called the Quasi Curvature-Dimension
condition $\mathsf{QCD}(Q,K,N)$, see [68, Def. 2.3]. $Q\geq 1$ is a an
auxiliary parameter so that for $Q=1$ the $\mathsf{QCD}(1,K,N)$ is equivalent
to the $\mathsf{CD}(K,N)$, and it is strictly weaker than the latter for
$Q>1$. With the same arguments used in the proof of Theorem 3.12, for
essentially non-branching m.m.s., the $\mathsf{QCD}(Q,K,N)$ for $N=n$
corresponds to the $\mathsf{CD}(\beta,n)$ inequality (3.16) with gauge
function $\mathsf{G}=\mathsf{d}$, and

$\beta_{t}(\theta)=\frac{1}{Q}(\beta^{\tau}_{K,n})_{t}(\theta),\qquad\forall\,t\in[0,1],\quad\forall\,\theta\in[0,+\infty),$
(3.43)

where $(\beta^{\tau}_{K,n})_{t}$ is the same coefficient (3.32) of the Lott-
Sturm-Villani’s theory. Formally the coefficient (3.43) is not of the form
(3.14) due to the extra factor $1/Q$. However, all results of Section 4 hold
with the same proofs for this more general class of coefficients, up to
keeping track of this extra factor in the statements.

On the other hand the finer Curvature-Geodesic-Topological-Dimension condition
$\mathsf{CGTD}(K,N,n)$ introduced by E. Milman in [68, Sec. 7], for
$K\in\mathbb{R}$, $n\geq 1$ and $N\geq n$ corresponds to the
$\mathsf{CD}(\beta_{K,N}^{\tau},n)$ for gauge function
$\mathsf{G}=\mathsf{d}$.

Notice that for $Q=1$ and $N=n$, $\mathsf{CGTD}(K,N,n)$ and
$\mathsf{QCD}(Q,K,N)$ are equivalent, and in turn equivalent to the classical
$\mathsf{CD}(K,N)$.

####  3.3.2 Recovering the Balogh-Kristàly-Sipos’ theory

In this section we show how the general theory introduced above also contains
the one developed in [19] for Heisenberg groups.

Let $d\geq 1$. The Heisenberg group is the non-commutative group structure on
$\mathbb{R}^{2d+1}$ given, in coordinates
$(x_{1},\ldots,x_{d},y_{1},\ldots,y_{d},z)$, by the law

$(x,y,z)\star(x^{\prime},y^{\prime},z^{\prime})=\left(x+x^{\prime},y+y^{\prime},z+z^{\prime}+\sum_{i=1}^{d}\frac{1}{2}(x_{i}y_{i}^{\prime}-y_{i}x_{i}^{\prime})\right).$
(3.44)

Consider the $2d$ left-invariant vector fields

$X_{i}=\frac{\partial}{\partial x_{i}}-\frac{y_{i}}{2}\frac{\partial}{\partial
z},\qquad Y_{i}=\frac{\partial}{\partial
y_{i}}+\frac{x_{i}}{2}\frac{\partial}{\partial z},\qquad i=1,\ldots,d,$ (3.45)

and the left-invariant metric making $\\{X_{i},Y_{i}\\}_{i=1}^{d}$ a global
orthonormal frame. We equip $\mathbb{H}^{d}$ with the associated sub-
Riemannian (Carnot-Carathéodory) distance, denoted by $\mathsf{d}_{cc}$ (see
Appendix A), and the Lebesgue measure $\mathscr{L}^{2d+1}$, which a Haar
measure.

We set

$\mathsf{s}(\theta)=\theta\sin\left(\frac{\theta}{2}\right)^{2d-1}\left[\sin\left(\frac{\theta}{2}\right)-\frac{\theta}{2}\cos\left(\frac{\theta}{2}\right)\right],\qquad\theta\in[0,+\infty).$
(3.46)

Notice that $\mathsf{s}(\theta)$ has order $N=2d+3$ as $\theta\to 0$.
Moreover, it holds

$\mathcal{D}=\inf\\{\theta>0\mid\mathsf{s}_{d}(\theta)=0\\}=2\pi.$ (3.47)

The corresponding distortion coefficient, following the recipe of (3.14), is
then

$[0,1]\times[0,+\infty]\ni(t,\theta)\mapsto\\\
\beta^{\mathbb{H}^{d}}_{t}(\theta)=\begin{cases}t^{2d+3}&\theta=0,\\\
t\dfrac{\sin\left(\frac{t\theta}{2}\right)^{2d-1}\left[\sin\left(\frac{t\theta}{2}\right)-\frac{t\theta}{2}\cos\left(\frac{t\theta}{2}\right)\right]}{\sin\left(\frac{\theta}{2}\right)^{2d-1}\left[\sin\left(\frac{\theta}{2}\right)-\frac{\theta}{2}\cos\left(\frac{\theta}{2}\right)\right]}&0<\theta<2\pi,\\\
+\infty&\theta\geq 2\pi,t\neq 0,\\\ 0&\theta\geq 2\pi,t=0.\\\ \end{cases}$
(3.48)

We highlight in particular the following identity

$\beta^{\mathbb{H}^{d}}_{t}(\theta)=(\tau_{t}^{d}(\theta))^{2d+1},\qquad\forall\,t\in[0,1],\quad\forall\,\theta\in[0,2\pi],$
(3.49)

where $\tau_{s}^{d}(\theta)$ is the function considered in [19, Eq. (1.6)].

The main result [19, Thm. 1.1] is a Jacobian determinant inequality which
permits by standard manipulations to obtain an interpolation inequality for
optimal transport, see [19, Eq. (3.17), p. 61]. More precisely, for all
$\mu_{0},\mu_{1}\in\mathcal{P}_{bs}^{*}(\mathbb{H}^{d},\mathsf{d}_{cc},\mathscr{L}^{2d+1})$,
there exists $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ associated with a
$W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ such that
$\mu_{t}\ll\mathscr{L}^{2d+1}$ for all $t\in(0,1]$ and for $\nu$-a.e. $\gamma$
it holds

$\frac{1}{\rho_{t}(\gamma_{t})^{1/(2d+1)}}\geq\frac{\beta^{\mathbb{H}^{d}}_{1-t}(\theta^{\gamma_{1},\gamma_{0}})^{1/(2d+1)}}{\rho_{0}(\gamma_{0})^{1/(2d+1)}}+\frac{\beta^{\mathbb{H}^{d}}_{t}(\theta^{\gamma_{0},\gamma_{1}})^{1/(2d+1)}}{\rho_{1}(\gamma_{1})^{1/(2d+1)}},\qquad\forall\,t\in[0,1],$
(3.50)

where $\rho_{t}=\frac{\mathrm{d}\mu_{t}}{\mathrm{d}\mathscr{L}^{2d+1}}$, and
$\theta^{x,y}$ is the vertical norm of the covector associated with the
geodesic joining $x$ and $y$. In (3.50) we use the convention that if
$\mu_{0}$ is not absolutely continuous then the first term in the right hand
side is omitted.

Using the definition of $\mathrm{U}_{n}$, the convexity of the map
$\mathbb{R}^{2}\ni(x,y)\mapsto\log(e^{x}+e^{y})$ and Jensen’s inequality, one
establishes the validity of the $\mathsf{CD}(\beta,2d+1)$, with
$\beta=\beta^{\mathbb{H}^{d}}$.

We stress that the argument in the r.h.s. of (3.50) does not depend on the
distance. It is then natural to set as a gauge function the map
$\mathsf{G}:\mathbb{H}^{d}\times\mathbb{H}^{d}\to[0,+\infty]$ by

$\mathsf{G}(x,y):=\theta^{x,y}.$ (3.51)

Then the results of [19] can be restated within our framework as follows.

######  Theorem 3.15.

The gauge m.m.s.
$(\mathbb{H}^{d},\mathsf{d}_{cc},\mathscr{L}^{2d+1},\mathsf{G})$ satisfies the
$\mathsf{CD}(\beta^{\mathbb{H}^{d}},2d+1)$.

It is not hard to check that $\beta^{\mathbb{H}^{d}}_{t}(\theta)\geq t^{2d+3}$
for all $t\in[0,1]$ and $\theta\in[0,2\pi]$. Therefore, by Proposition
3.10(iii), the $\mathsf{CD}(\beta^{\mathbb{H}^{d}},2d+1)$ implies the
$\mathsf{CD}(t^{2d+3},2d+1)$. In turn, the latter implies the classical
$\mathsf{MCP}(0,2d+3)$, originally established for $\mathbb{H}^{d}$ by Juillet
in [53].

##  4 Geometric consequences

In this section we establish some geometric properties implied by the
$\mathsf{CD}(\beta,n)$ condition (compare with [44, 82, 64] for the Lott-
Sturm-Villani’s $\mathsf{CD}(K,N)$ spaces).

###  4.1 Generalized Brunn-Minkowski inequality

Given two Borel sets $A_{0},A_{1}\subset X$, denote with $A_{t}$ the set of
$t$-intermediate points, i.e.

$A_{t}:=Z_{t}(A_{0},A_{1})=\\{\gamma_{t}\mid\gamma\in\mathrm{Geo}(X),\,\gamma_{0}\in
A_{0},\,\gamma_{1}\in A_{1}\\}.$ (4.1)

It is well-known that $A_{t}$ is a Suslin set. For non-empty Borel sets
$A_{0},A_{1}\subset X$, denote

$\beta_{t}(A_{0},A_{1}):=\sup_{\tilde{A}_{i}\subseteq
A_{i}}\inf\\{\beta_{t}(\mathsf{G}(x_{0},x_{1}))\mid
x_{i}\in\tilde{A}_{i}\cap\operatorname{\mathrm{supp}}\mathfrak{m}\\},$ (4.2)

where the sup is taken over all full-measure non-empty subsets
$\tilde{A}_{i}\subseteq A_{i}$, for $i=0,1$. It is clear from the definition
(4.2) that

$\mathfrak{m}(A_{0}\triangle\tilde{A}_{0})=\mathfrak{m}(A_{1}\triangle\tilde{A}_{1})=0\quad\Longrightarrow\quad\beta_{t}(A_{0},A_{1})=\beta_{t}(\tilde{A}_{0},\tilde{A}_{1})\,,$
(4.3)

where $A_{0},\tilde{A}_{0},A_{1},\tilde{A}_{1}\subset X$ are Borel sets and
$\triangle$ denotes the symmetric difference of sets.

######  Theorem 4.1 (Generalized Brunn-Minkowski inequality).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$, satisfying the $\mathsf{CD}(\beta,n)$ condition, with
$n\in[1,+\infty)$ and $\beta$ as in (3.14). Let $A_{0},A_{1}\subset X$ be
Borel sets with $\mathfrak{m}(A_{0}),\mathfrak{m}(A_{1})>0$ and
$\mathfrak{m}(A_{0}\cap A_{1})=0$. Then

$\bar{\mathfrak{m}}(A_{t})^{1/n}\geq\beta_{1-t}(A_{1},A_{0})^{1/n}\cdot\mathfrak{m}(A_{0})^{1/n}+\beta_{t}(A_{0},A_{1})^{1/n}\cdot\mathfrak{m}(A_{1})^{1/n},\qquad\forall\,t\in(0,1),$
(4.4)

where $\bar{\mathfrak{m}}$ is the completion of $\mathfrak{m}$, $A_{t}$ is the
set of $t$-intermediate points (4.1), and with the convention that
$0\cdot\infty=0$.

###### Proof.

Assume that $\mathfrak{m}(A_{i})<+\infty$ for $i=0,1$. We first reduce the
proof to the bounded and disjoint case. Since $\mathfrak{m}(A_{i})<+\infty$
and $\mathfrak{m}(A_{0}\cap A_{1})=0$, for any $\varepsilon>0$ there exists
$A_{i,\varepsilon}\subset A_{i}$ which is closed and bounded,
$A_{0,\varepsilon}\cap A_{1,\varepsilon}=\emptyset$, and
$\mathfrak{m}(A_{i,\varepsilon})\to\mathfrak{m}(A_{i})$ as $\varepsilon\to 0$.
Furthermore, setting
$A_{t,\varepsilon}:=Z_{t}(A_{0,\varepsilon},A_{1,\varepsilon})$ the set of
$t$-intermediate points from $A_{0,\varepsilon}$ to $A_{1,\varepsilon}$, it
holds

$\beta_{t}(A_{0}^{\varepsilon},A_{1}^{\varepsilon})\geq\beta_{t}(A_{0},A_{1}),\qquad\beta_{t}(A_{1}^{\varepsilon},A_{0}^{\varepsilon})\geq\beta_{t}(A_{1},A_{0}),\qquad
A_{t}\supset A_{t}^{\varepsilon}.$ (4.5)

Thus it is sufficient to prove (4.4) for closed and bounded sets $A_{0},A_{1}$
with $A_{0}\cap A_{1}=\emptyset$. Let
$\mu_{i}:=\tfrac{1}{\mathfrak{m}(A_{i})}\mathfrak{m}\llcorner_{A_{i}}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$
for $i=0,1$ and note that
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$.
Observe that $\mathrm{U}_{n}(\mu_{i})=\mathfrak{m}(A_{i})^{1/n}$. By
assumption, there exists a $W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ joining
them, and satisfying the inequality (3.15). We remark that $\mu_{t}$ is
concentrated on $A_{t}$, which is a Suslin set. Using Jensen’s inequality
twice, we obtain

$\mathrm{U}_{n}(\mu_{t})=\exp\left(-\frac{1}{n}\int_{X}\log\,\rho_{t}\,\mathrm{d}\mu_{t}\right)\leq\int_{X}\rho_{t}^{-1/n}\,\mathrm{d}\mu_{t}=\int_{A_{t}}\rho_{t}^{1-\frac{1}{n}}\mathrm{d}\bar{\mathfrak{m}}\leq\bar{\mathfrak{m}}(A_{t})^{1/n}.$
(4.6)

Then (4.4) for sets of finite measure follows directly from the
$\mathsf{CD}(\beta,n)$ inequality (3.15), and from optimizing with respect to
full-measure subsets of $A_{i}$.

Since $\mathfrak{m}$ is $\sigma$-finite, we can prove the result for sets of
possibly infinite measure by approximating them with a monotone sequence of
sets with finite measure. ∎

With analogous arguments one obtains the next generalized half-Brunn-Minkowski
inequality for $\mathsf{MCP}(\beta)$, which will play a pivotal role in
geometric applications.

######  Theorem 4.2 (Generalized half-Brunn-Minkowski inequality).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$, satisfying the $\mathsf{MCP}(\beta)$ condition, with $\beta$ as
in (3.14). Let $A$ be a Borel set with $\mathfrak{m}(A)>0$, and
$\bar{x}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ with
$\mathfrak{m}(\\{\bar{x}\\}\cap A)=0$. Then

$\bar{\mathfrak{m}}(A_{t})\geq\beta_{t}(\bar{x},A)\cdot\mathfrak{m}(A),\qquad\forall\,t\in(0,1),$
(4.7)

where $\bar{\mathfrak{m}}$ is the completion of $\mathfrak{m}$, $A_{t}$
denotes the set of $t$-intermediate points (4.1) between $A_{0}=\\{\bar{x}\\}$
and $A_{1}=A$, and with the convention that $0\cdot\infty=0$.

######  Remark 4.3.

In case $\operatorname{\mathrm{supp}}\mathfrak{m}$ is not a singleton, the
condition $\mathfrak{m}(\\{x\\})=0$ for every $x\in X$ is very natural: indeed
it is satisfied for all smooth measures typically employed in sub-Riemannian
geometry, and it is implied by the generalized Bishop-Gromov Theorem 4.24(ii)
under the assumption that $\mathsf{G}$ is meek (see Definition 4.22) and that
the natural assumption (4.54) holds (see also Remark 4.25). Recall that
$\mathsf{MCP}(K,N)$ spaces (for $N\in[1,+\infty)$, $K\in\mathbb{R}$), thus
including all $\mathsf{CD}(K,N)$ ones, satisfy this condition.

######  Remark 4.4.

Let $(\mu_{t})_{t\in[0,1]}$ be a $W_{2}$-geodesic between the characteristic
measures of $A_{0}$ and $A_{1}$, for which the $\mathsf{CD}(\beta,n)$
inequality (3.15) holds. Then in the l.h.s. of (4.4) one can replace $A_{t}$
with any (possibly smaller) measurable set on which $\mu_{t}$ is concentrated.
An analogous remark applies for the l.h.s. of (4.7).

If the distortion coefficients are non-increasing or non-decreasing, then the
above inequalities can be expressed in terms of the minimal/maximal value of
the gauge function $\mathsf{G}$ on those set. More precisely, if
$\theta\mapsto\beta_{t}(\theta)$ is non-decreasing or non-increasing for all
$t\in(0,1)$, then letting

$\Theta:=\begin{cases}\inf\\{\mathsf{G}(x_{0},x_{1})\mid x_{i}\in
A_{i}\cap\operatorname{\mathrm{supp}}\mathfrak{m},\;i=1,2\\}&\text{if
$\theta\mapsto\beta_{t}(\theta)$ is non-decreasing},\\\
\sup\\{\mathsf{G}(x_{0},x_{1})\mid x_{i}\in
A_{i}\cap\operatorname{\mathrm{supp}}\mathfrak{m},\;i=1,2\\}&\text{if
$\theta\mapsto\beta_{t}(\theta)$ is non-increasing},\end{cases}$ (4.8)

it holds

$\beta_{t}(A_{0},A_{1})\geq\beta_{t}(\Theta).$ (4.9)

This observation, together with the fact that the classical distortion
coefficients

$(\beta_{K,N}^{\tau})_{t}(\theta)=\tau_{K,N}^{(t)}(\theta)^{N},\qquad\text{resp.}\qquad(\beta_{K,N}^{\sigma})_{t}(\theta)=\sigma_{K/N}^{(t)}(\theta)^{N},$
(4.10)

are non-increasing or non-decreasing (according to the sign of $K$), yield the
familiar forms of the generalized Brunn-Minkowski inequalities for classical
$\mathsf{CD}(K,N)$ spaces [82] (resp. for classical $\mathsf{CD}^{*}(K,N)$ or
$\mathsf{CD}^{e}(K,N)$ spaces [17, 44]; see also Theorem 3.12 and Remark
3.13).

###  4.2 MCP implies CD for spaces supporting interpolation inequalities

The implication $\mathsf{CD}\Rightarrow\mathsf{MCP}$ is trivial with our
definitions. Here we discuss the converse implication. The result of this
section can be applied to ideal sub-Riemannian manifolds of topological
dimension $n$, that support interpolation inequalities for densities as in
[24].

######  Definition 4.5.

We say that a m.m.s. $(X,\mathsf{d},\mathfrak{m})$ _supports interpolation
inequalities for densities, with dimensional parameter $n\in[1,+\infty)$_, if
for any $\mu_{0}\in\mathcal{P}_{bs}(X,\mathsf{d},\mathfrak{m})$,
$\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$ with
$\operatorname{\mathrm{supp}}\mu_{0}\cap\operatorname{\mathrm{supp}}\mu_{1}=\emptyset$
there exists a $W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$, induced by
$\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ such that $\mu_{t}\ll\mathfrak{m}$ for
all $t\in(0,1]$, and letting
$\rho_{t}:=\tfrac{\mathrm{d}\mu_{t}}{\mathrm{d}\mathfrak{m}}$ be the
corresponding density, it holds for all $t\in(0,1)$:

$\frac{1}{\rho_{t}(\gamma_{t})^{1/n}}\geq\frac{\beta_{1-t}^{(X,\mathsf{d},\mathfrak{m})}(\gamma_{1},\gamma_{0})^{1/n}}{\rho_{0}(\gamma_{0})^{1/n}}+\frac{\beta_{t}^{(X,\mathsf{d},\mathfrak{m})}(\gamma_{0},\gamma_{1})^{1/n}}{\rho_{1}(\gamma_{1})^{1/n}},\qquad\nu\text{-a.e.}\
\gamma\in\mathrm{Geo}(X),$ (4.11)

with the understanding that the first term in the right hand side of (4.11) is
omitted if $\mu_{0}\notin\mathcal{P}_{ac}(X,\mathfrak{m})$.

In (4.11), $\beta_{t}^{(X,\mathsf{d},\mathfrak{m})}$ denotes the “true”
distortion coefficient of the metric measure space
$(X,\mathsf{d},\mathfrak{m})$, of which we recall the definition.

######  Definition 4.6.

The _distortion coefficient_ of $(X,\mathsf{d},\mathfrak{m})$ is the map
$\beta_{(\cdot)}^{(X,\mathsf{d},\mathfrak{m})}(\cdot,\cdot):[0,1]\times
X\times X\to[0,+\infty]$ defined by

$\beta_{t}^{(X,\mathsf{d},\mathfrak{m})}(x,y):=\limsup_{r\to
0^{+}}\frac{\bar{\mathfrak{m}}(Z_{t}(\\{x\\},B_{r}(y)))}{\mathfrak{m}(B_{r}(y))},\qquad\forall\,x,y\in
X,\quad t\in[0,1],$ (4.12)

where $Z_{t}(A,B)$ denotes the set of $t$-midpoints between two Borel sets as
in (4.1), and $\bar{\mathfrak{m}}$ denotes the completion of $\mathfrak{m}$.

######  Theorem 4.7 ($\mathsf{MCP}$ to $\mathsf{CD}$).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. that supports interpolation
inequalities for densities, with dimensional parameter $n\in[1,+\infty)$. Let
$\mathsf{G}:X\times X\to[0,+\infty]$ be a gauge function. Assume that:

  * •

each $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ occurring in Definition 4.5 is
concentrated on a set of geodesics whose endpoints $(\gamma_{0},\gamma_{1})$
and $(\gamma_{1},\gamma_{0})$ are continuity points for $\mathsf{G}$;

  * •

$(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies the $\mathsf{MCP}(\beta)$,
with $\beta$ as in (3.14), and with $N\geq n$.

Then $(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ also satisfies the
$\mathsf{CD}(\beta,n)$.

###### Proof.

Let $x\neq y\in\operatorname{\mathrm{supp}}\mathfrak{m}$ with $(x,y)$ a
continuity point of $\mathsf{G}$, and let $r<\mathsf{d}(x,y)$. We use the
half-Brunn-Minkowski inequality for $\mathsf{MCP}(\beta)$ spaces of
Proposition 4.2, for $\bar{x}=x$, $A=B_{r}(y)$. It holds, for all $t\in[0,1]$,

$\frac{\bar{\mathfrak{m}}(Z_{t}(\\{x\\},B_{r}(y)))}{\mathfrak{m}(B_{r}(y))}\geq\beta_{t}(x,B_{r}(y))\geq\inf\\{\beta_{t}(\mathsf{G}(x,z))\mid
z\in B_{r}(y)\\}.$ (4.13)

Taking the $\limsup$ for $r\to 0^{+}$, and using the continuity of
$\mathsf{G}$ at $(x,y)$, we find a sequence
$\theta_{j}\to\mathsf{G}(x,y)\in[0,+\infty]$ such that

$\beta_{t}^{(X,\mathsf{d},\mathfrak{m})}(x,y)\geq\liminf_{j}\beta_{t}(\theta_{j})=\beta_{t}(\mathsf{G}(x,y)),\qquad\forall\,t\in[0,1].$
(4.14)

For the latter equality, we used Proposition 3.4 (indeed, if
$\mathsf{G}(x,y)<+\infty$ then $\liminf_{j}$ is a actually a $\lim_{j}$, but
if $\mathsf{G}(x,y)=\mathcal{D}=+\infty$ then we only have lower semi-
continuity).

Let $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ be as in Definition 4.5. Assume for
simplicity that $\mu_{0}\in\mathcal{P}_{ac}(X,\mathfrak{m})$ (the argument
works verbatim if $\mu_{0}\notin\mathcal{P}_{ac}(X,\mathfrak{m})$ by omitting
the corresponding term in all inequalities). By our assumption, $\nu$ is
concentrated on a set of geodesics $\gamma$ such that $\mathsf{G}$ is
continuous at $(\gamma_{0},\gamma_{1})$ and $(\gamma_{1},\gamma_{0})$.

For $\nu$-a.e. $\gamma$ we can plug (4.14) in (4.11), finding that for all
$t\in(0,1)$ it holds

$\frac{1}{\rho_{t}(\gamma_{t})^{1/n}}\geq\frac{\beta_{1-t}(\mathsf{G}(\gamma_{1},\gamma_{0}))^{1/n}}{\rho_{0}(\gamma_{0})^{1/n}}+\frac{\beta_{t}(\mathsf{G}(\gamma_{0},\gamma_{1}))^{1/n}}{\rho_{1}(\gamma_{1})^{1/n}},\qquad\nu\text{-a.e.}\
\gamma.$ (4.15)

Using the definition of $\mathrm{U}_{n}$, the convexity of the map
$\mathbb{R}^{2}\ni(x,y)\mapsto\log(e^{x}+e^{y})$ and Jensen’s inequality, we
obtain the $\mathsf{CD}(\beta,n)$ inequality. ∎

###  4.3 Gauge diameter estimate

######  Definition 4.8 ($\mathsf{G}$-diameter).

The _$\mathsf{G}$ -diameter_ of a non-empty set $S$ is defined by:

$\operatorname{\mathrm{diam}}_{\mathsf{G}}(S)=\sup_{x\in
S}\left(\mathfrak{m}\shortminus\operatorname*{ess\,sup}_{y\in S,\,y\neq
x}\mathsf{G}(x,y)\right).$ (4.16)

We say that $S\subset X$ is _$\mathsf{G}$ -bounded_ if is non-empty and it has
finite $\mathsf{G}$-diameter.

Recall that the parameter $\mathcal{D}$ in (3.13), when finite, characterizes
the blow-up of $\theta\mapsto\beta_{t}(\theta)$, for $t\in(0,1)$.

######  Proposition 4.9 ($\mathsf{G}$-diameter estimate).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$. Let $\beta$ be as in (3.14) and assume that
$\beta_{t}(+\infty)=+\infty$ for all $t\in(0,1)$. Note this is in particular
the case if $\mathcal{D}<+\infty$.

  * •

If $(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies the
$\mathsf{MCP}(\beta)$, then for all
$x\in\operatorname{\mathrm{supp}}\mathfrak{m}$ it holds

$\mathsf{G}(x,y)<\mathcal{D},\qquad\mathfrak{m}\text{-a.e. }y\in X,\quad y\neq
x.$ (4.17)

In particular,
$\operatorname{\mathrm{diam}}_{\mathsf{G}}(\operatorname{\mathrm{supp}}\mathfrak{m})\leq\mathcal{D}$.

  * •

If $(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies the
$\mathsf{CD}(\beta,n)$ for some $n\in[1,+\infty)$, then

$\mathsf{G}(x,y)<\mathcal{D},\qquad\pi\text{-a.e. }(x,y)\in X\times X.$ (4.18)

for all $\pi=({\rm e}_{0},{\rm e}_{1})_{\sharp}\nu$, where $\nu\in{\rm
OptGeo}(\mu_{0},\mu_{1})$ is such that inequality (3.15) holds.

######  Remark 4.10.

We remark that the case $\mathcal{D}=+\infty$ is included in the statement of
Proposition 4.9. In this case, $\mathfrak{m}(\\{y\in X\mid y\neq
x,\,\mathsf{G}(x,y)=+\infty\\})=0$ for all
$x\in\operatorname{\mathrm{supp}}\mathfrak{m}$. This implies the non-
triviality condition (4.41) considered below.

###### Proof.

Fix $x\in\operatorname{\mathrm{supp}}\mathfrak{m}$. We show that the set
$A=\\{y\in X\mid y\neq x,\,\mathsf{G}(x,y)\geq\mathcal{D}\\}\cap B_{R}(x)$ has
zero measure for all $R>0$. By contradiction, assume that $\mathfrak{m}(A)>0$.
Then we apply the half-Brunn-Minkowski inequality of Theorem 4.2 to the set
$A$. Notice that $\beta_{t}(x,A)=+\infty$ for all $t\in(0,1)$, and that
$A_{t}\subset(B_{R}(x))_{t}\subset B_{tR}(x)$. We obtain hence

$\mathfrak{m}(B_{tR}(x))\geq\bar{\mathfrak{m}}(A_{t})=+\infty,\qquad\forall\,t\in(0,1),$
(4.19)

contradicting the local finiteness of $\mathfrak{m}$.

To prove the second part of the proposition, we argue in a similar way: let
$\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ as in the statement (in particular
$\mu_{0},\mu_{1}$ have bounded support) such that $\pi(\\{(x,y)\in X\times
X\mid\mathsf{G}(x,y)\geq\mathcal{D}\\})>0$, and using (3.15) we obtain that,
for the corresponding $W_{2}$-geodesic $\mu_{t}=({\rm e}_{t})_{\sharp}\nu$ it
holds

$\mathrm{U}_{n}(\mu_{t}|\mathfrak{m})=+\infty,\qquad\forall\,t\in(0,1).$
(4.20)

This implies that $\mathfrak{m}(\operatorname{\mathrm{supp}}\mu_{t})=+\infty$.
However, since $\operatorname{\mathrm{supp}}\mu_{0}$ and
$\operatorname{\mathrm{supp}}\mu_{1}$ are bounded, then also
$\operatorname{\mathrm{supp}}\mu_{t}$ is bounded, yielding a contradiction. ∎

For some interesting cases (cf. Section 7), it holds
$\mathsf{G}\geq\mathsf{d}$. In this case, Proposition 4.9 admits the following
Bonnet-Myers type result.

######  Corollary 4.11 (Bonnet-Myers).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$, satisfying the $\mathsf{MCP}(\beta)$ with $\beta$ as in (3.14).
Assume that $\mathsf{G}\geq\mathsf{d}$. Then
$\operatorname{\mathrm{diam}}(\operatorname{\mathrm{supp}}\mathfrak{m})\leq\mathcal{D}$,
and if $\mathcal{D}<+\infty$ then $\operatorname{\mathrm{supp}}\mathfrak{m}$
is compact.

###### Proof.

The estimate on the diameter is immediate by continuity of $\mathsf{d}$. If
$\mathcal{D}<+\infty$, then $\operatorname{\mathrm{supp}}\mathfrak{m}$ is
bounded, and by Proposition 4.9 it is also $\mathsf{G}$-bounded. By Corollary
4.15, $\operatorname{\mathrm{supp}}\mathfrak{m}$ is totally bounded, complete,
and thus compact. ∎

###  4.4 Doubling inequalities for balls

We first prove a local doubling inequality for metric balls under minimal
assumptions. We stress that “locality” in Proposition 4.12 is measured with
respect to the gauge function, which also determines the doubling constant.
For this reason, and without additional relations between $\mathsf{G}$ and
$\mathsf{d}$, the general form of the statement is subtly different with
respect to the classical one.

######  Proposition 4.12 (Local doubling).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$, satisfying the $\mathsf{MCP}(\beta)$ with $\beta$ as in (3.14).
Then any $\mathsf{G}$-bounded Borel subset
$S\subseteq\operatorname{\mathrm{supp}}\mathfrak{m}$ satisfies the following
inequality: for all $t\in(0,1)$ there exists $C_{S,t}>0$ such that

$\mathfrak{m}(B_{r}(x_{0})\cap S)\leq
C_{S,t}\cdot\mathfrak{m}(B_{tr}(x_{0})),\qquad\forall\,r\geq
0,\;\forall\,x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}.$ (4.21)

The constant $C_{S,t}$ can be estimated in terms of $\beta$ and
$\operatorname{\mathrm{diam}}_{\mathsf{G}}(S)$:

$\frac{1}{C_{S,t}}=\inf\\{\beta_{t}(\theta)\mid\theta\in[0,\operatorname{\mathrm{diam}}_{\mathsf{G}}(S)]\\}\in(0,t^{N}].$
(4.22)

###### Proof.

Fix $S\subseteq\operatorname{\mathrm{supp}}\mathfrak{m}$ and let
$\mathsf{G}_{0}:=\operatorname{\mathrm{diam}}_{\mathsf{G}}(S)<\infty$. We let

$\frac{1}{C_{S,t}}:=\inf\\{\beta_{t}(\theta)\mid\theta\in[0,\mathsf{G}_{0}]\\}\in(0,t^{N}].$
(4.23)

Here, the upper bound is a consequence of the value of $\beta_{t}(0)=t^{N}$ in
(3.14), while the lower bound follows from 3.4(vi). The case $r=0$ of (4.21)
is trivial, so let $r>0$.

Let $x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$, and let
$\check{B}_{r}(x_{0}):=B_{r}(x_{0})\setminus\\{x_{0}\\}$. Let
$A=\check{B}_{r}(x_{0})\cap S$. If $\mathfrak{m}(A)=0$ there is nothing to
prove. Assume then that $\mathfrak{m}(A)>0$, and note that
$\mathfrak{m}(A\cap\\{x_{0}\\})=0$. Thus we can apply the half-Brunn-Minkowski
estimate of Theorem 4.2 with $\bar{x}=x_{0}$ and $A=\check{B}_{r}(x_{0})\cap
S$. Since $A_{t}\subseteq\check{B}_{tr}(x_{0})$, we have for all $t\in(0,1)$:

$\mathfrak{m}(\check{B}_{tr}(x_{0}))\geq\beta_{t}(x_{0},\check{B}_{r}(x_{0})\cap
S)\cdot\mathfrak{m}(\check{B}_{r}(x_{0})\cap
S)\geq\tfrac{1}{C_{S,t}}\cdot\mathfrak{m}(\check{B}_{r}(x_{0})\cap S).$ (4.24)

We justify the bound $\beta_{t}(x_{0},\check{B}_{r}(x_{0})\cap S)\geq
1/C_{S,t}$. To do it, for any $x\in\operatorname{\mathrm{supp}}\mathfrak{m}$
and any non-empty Borel set $U\subseteq S$, recall the definition (4.2), which
for our case reduces to

$\beta_{t}(x,U)=\sup_{\tilde{U}\subseteq
U}\inf\\{\beta_{t}(\mathsf{G}(x,y))\mid y\in\tilde{U}\\},$ (4.25)

where $\tilde{U}$ ranges over all full-measure and non-empty subsets of $U$.
Assume that $x\notin U$. Since $U\subseteq S$ and the latter has bounded
$\mathsf{G}$-diameter (see Definition 4.8), there exists a full-measure set
$U^{\prime}\subseteq U$ such that $\mathsf{G}(x,y)\leq\mathsf{G}_{0}$ for all
$y\in U^{\prime}$, and in particular it holds

$\beta_{t}(x,U)\geq\inf\\{\beta_{t}(\theta)\mid\theta\in[0,\mathsf{G}_{0}]\\}=\frac{1}{C_{S,t}}.$
(4.26)

In particular the above inequality holds for $x=x_{0}$ and
$U=\check{B}_{r}(x_{0})$ as required.

Thus we have proved (4.21) for $\check{B}_{r}(x_{0})$ in place of
$B_{r}(x_{0})$, that is

$\mathfrak{m}(\check{B}_{r}(x_{0})\cap S)\leq
C_{S,t}\cdot\mathfrak{m}(\check{B}_{tr}(x_{0})),\quad\forall\,r\geq
0,\;\forall\,x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}.$ (4.27)

Since $C_{S,t}\geq 1/t^{N}\geq 1$, we add $\mathfrak{m}(\\{x_{0}\\}\cap S)$ to
both sides to obtain (4.21). ∎

In some natural applications, bounded sets are $\mathsf{G}$-bounded, cf.
Section 7. In this case Proposition 4.12 can be restated as a classical
doubling inequality.

######  Corollary 4.13 (Classical local doubling).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$, satisfying the $\mathsf{MCP}(\beta)$ with $\beta$ as in (3.14).
Assume that bounded subsets of $\operatorname{\mathrm{supp}}\mathfrak{m}$ are
$\mathsf{G}$-bounded. Then for any bounded set
$S\subseteq\operatorname{\mathrm{supp}}\mathfrak{m}$ and $R>0$ there exists
$C=C(S,R)>0$ such that

$\mathfrak{m}(B_{2r}(x_{0}))\leq
C\cdot\mathfrak{m}(B_{r}(x_{0})),\qquad\forall\,r\in[0,R],\;\forall\,x_{0}\in
S.$ (4.28)

###### Proof.

Let $R_{0}$ be the diameter of $S$, and let
$S^{\prime}:=B_{2R+R_{0}}(\bar{x})$ be a ball centered in any point
$\bar{x}\in S$. Of course any ball $B_{2r}(x_{0})$ with $r\in[0,R]$ and
$x_{0}\in S$ is contained in $S^{\prime}$. By our additional hypothesis,
$S^{\prime}$ is also $\mathsf{G}$-bounded. We apply then Proposition 4.12 to
the set $S^{\prime}$ with $t=1/2$ and get the claim with
$C=C_{S^{\prime},1/2}$. ∎

Finally, if
$\operatorname{\mathrm{diam}}_{\mathsf{G}}(\operatorname{\mathrm{supp}}\mathfrak{m})$
is finite, or if $\theta\mapsto\beta_{t}(\theta)$ is non-decreasing, then we
have a classical global doubling inequality.

######  Corollary 4.14 (Classical global doubling).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$, satisfying the $\mathsf{MCP}(\beta)$ with $\beta$ as in (3.14).
Assume that
$\operatorname{\mathrm{diam}}_{\mathsf{G}}(\operatorname{\mathrm{supp}}\mathfrak{m})$
is finite, or that $\theta\mapsto\beta_{t}(\theta)$ is non-decreasing. Then
there exists a constant $C>0$ such that

$\mathfrak{m}(B_{2r}(x_{0}))\leq
C\cdot\mathfrak{m}(B_{r}(x_{0})),\qquad\forall\,r\in[0,+\infty),\;\forall\,x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}.$
(4.29)

###### Proof.

In both cases, using the same argument of the proof of Proposition 4.12,
taking $S=\operatorname{\mathrm{supp}}\mathfrak{m}$, we have that

$\beta_{t}(x_{0},B_{r}(x_{0}))\geq\frac{1}{C(t)}>0,$ (4.30)

for a constant $C(t)>0$ that does not depend on
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ or $r\in[0,+\infty)$: if
$\operatorname{\mathrm{diam}}_{\mathsf{G}}(\operatorname{\mathrm{supp}}\mathfrak{m})$
is finite (4.30) follows from the definition of
$\beta_{t}(x_{0},B_{r}(x_{0}))$ and Proposition 3.4(vi), while if
$\theta\mapsto\beta_{t}(\theta)$ is non-decreasing the lower bound is given by
$\beta_{t}(0)=t^{N}$. ∎

We also record a standard consequence that will be useful in the following.

######  Corollary 4.15 (Proper geodesic space).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function
$\mathsf{G}$, satisfying the $\mathsf{MCP}(\beta)$ with $\beta$ as in (3.14).
Assume that bounded subsets of $\operatorname{\mathrm{supp}}\mathfrak{m}$ are
$\mathsf{G}$-bounded, or $\theta\mapsto\beta_{t}(\theta)$ is non-decreasing.
Then $(\operatorname{\mathrm{supp}}\mathfrak{m},\mathsf{d})$ is a proper,
geodesic space.

###### Proof.

The proof follows the classical strategy. Assume first that bounded subsets of
$\operatorname{\mathrm{supp}}\mathfrak{m}$ are $\mathsf{G}$-bounded. Let
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ and $L>0$. Let
$x_{1},\dots,x_{k}$ be a collection of $k\in\mathbb{N}$ points in
$B_{L}(x_{0})$, and $\varepsilon>0$. If $B_{\varepsilon}(x_{i})\subseteq
B_{L}(x_{0})$ for all $i=1,\dots,k$ are pairwise disjoint, it holds

$\mathfrak{m}(B_{L}(x_{0}))\geq\sum_{i=1}^{k}\mathfrak{m}(B_{\varepsilon}(x_{i}))\geq\frac{1}{C_{S,t}}\sum_{i=1}^{k}\mathfrak{m}(B_{2L}(x_{i})\cap
S)\geq\frac{k}{C_{S,t}}\mathfrak{m}(B_{L}(x_{0})),$ (4.31)

where we applied Proposition 4.12 with $S=B_{3L}(x_{0})$ and
$t=\varepsilon/2L$. Thus, the maximal number of pairwise disjoint balls of
radius $\varepsilon>0$ contained in $B_{L}(x_{0})$ is bounded by

$k\leq C_{S,t}<+\infty,\qquad\text{with}\quad S=B_{3L}(x_{0}),\quad
t=\varepsilon/2L,$ (4.32)

where $C_{S,t}$ is the constant of (4.22). As a consequence, the ball
$B_{L}(x_{0})$ can be covered by $C_{S,t}$ metric balls of radius
$2\varepsilon$. Since $x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ and
$L>0$ were arbitrary, we obtain that balls in
$\operatorname{\mathrm{supp}}\mathfrak{m}$ are totally bounded, hence their
closure is compact. In other words the metric space
$(\operatorname{\mathrm{supp}}\mathfrak{m},\mathsf{d})$ is proper. Since
$(\operatorname{\mathrm{supp}}\mathfrak{m},\mathsf{d})$ is a length space by
the $\mathsf{MCP}(\beta)$ condition (cf. Remark 3.9), it is also geodesic.

If $\theta\mapsto\beta_{t}(\theta)$ is non-decreasing, the result is obtained
by the same argument but using, instead, the classical global doubling
inequality of Corollary 4.14. ∎

######  Remark 4.16.

We notice that as a by-product of the proof of Corollary 4.15, any
$\mathsf{G}$-bounded metric ball $B_{L}(x_{0})$ with
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ can be covered by a finite
number of balls of radius $\varepsilon$, and such a number depends only on
$\varepsilon$ and the $\mathsf{G}$-diameter of $B_{3L}(x_{0})$.

###  4.5 Geodesic dimension estimates

The parameter $N\in[1,+\infty)$ occurring in the definition of $\beta$ as in
(3.12) turns out to be a sharp upper bound for a different notion of
dimension, called _geodesic dimension_. The latter was introduced in [3] for
sub-Riemannian manifolds, and extended to metric measure spaces in [77]. We
recall its definition.

######  Definition 4.17 (Geodesic dimension).

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space. For any $x\in X$
and $s>0$, define

$C_{s}(x):=\sup\left\\{\limsup_{t\to
0^{+}}\frac{1}{t^{s}}\frac{\bar{\mathfrak{m}}(A_{t})}{\mathfrak{m}(A_{1})}\mid\text{$A_{1}$
Borel,\, bounded, $x\notin A_{1}$,\,
$\mathfrak{m}(A_{1})\in(0,+\infty)$}\right\\},$ (4.33)

where $A_{t}$ is the set of $t$-intermediate points between $A_{0}=\\{x\\}$
and $A_{1}$. We define the _geodesic dimension_ at $x\in X$ of
$(X,\mathsf{d},\mathfrak{m})$ as the non-negative real number

$\dim_{\mathrm{geo}}(x):=\inf\\{s>0\mid C_{s}(x)=+\infty\\}=\sup\\{s>0\mid
C_{s}(x)=0\\}\,,$ (4.34)

with the convention $\inf\emptyset=+\infty$ and $\sup\emptyset=0$. Finally,
for any subset $S\subset X$, we denote

$\dim_{\mathrm{geo}}(S):=\sup\\{\dim_{\mathrm{geo}}(x)\mid x\in S\\}.$ (4.35)

We remark that if $x\notin\operatorname{\mathrm{supp}}\mathfrak{m}$ we have
$\dim_{\mathrm{geo}}(x)=+\infty$ so it makes sense to consider only the
geodesic dimension at points in the support of $\mathfrak{m}$.

######  Lemma 4.18.

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. Then for any Borel set $S\subset
X$ it holds

$\dim_{\mathrm{geo}}(S)\geq\dim_{\mathrm{Haus}}(S).$ (4.36)

###### Proof.

We first observe that if there exists $z\in S$ with
$z\notin\operatorname{\mathrm{supp}}\mathfrak{m}$, we have
$\dim_{\mathrm{geo}}(z)=+\infty$, so that there is nothing to prove. Thus, we
assume $S\subseteq\operatorname{\mathrm{supp}}\mathfrak{m}$.

Let $S_{0}\subseteq S$ be any Borel subset of $S$ with
$\mathfrak{m}(S_{0})<+\infty$. We claim that there exists $x\in S_{0}$ such
that $\dim_{\mathrm{geo}}(x)\geq\dim_{\mathrm{Haus}}(S_{0})$. If
$\dim_{\mathrm{Haus}}(S_{0})=0$ the claim is true, so assume that
$\dim_{\mathrm{Haus}}(S_{0})>0$. If $k<\dim_{\mathrm{Haus}}(S_{0})$, then for
the $k$-dimensional Hausdorff measure it holds
$\mathcal{H}^{k}(S_{0})=+\infty$, so that by using [16, Thm. 2.4.3] there
exists $x\in S_{0}$ such that

$\limsup_{r\to 0^{+}}\frac{\mathfrak{m}(B_{r}(x))}{r^{k}}<+\infty.$ (4.37)

Let $A_{1}$ be a bounded Borel set not containing $x$ with
$\mathfrak{m}(A_{1})\subset(0,+\infty)$, and consider the corresponding set of
$t$-midpoints between $A_{1}$ and $A_{0}=\\{x\\}$. We have $A_{1}\subseteq
B_{R}(x)$ for some large $R>0$, so that $A_{t}\subseteq B_{tR}(x)$. Hence for
all $\varepsilon>0$ it holds

$\limsup_{t\to
0^{+}}\frac{1}{t^{k-\varepsilon}}\frac{\bar{\mathfrak{m}}(A_{t})}{\mathfrak{m}(A_{1})}\leq\limsup_{t\to
0^{+}}\frac{1}{t^{k-\varepsilon}}\frac{\mathfrak{m}(B_{tR}(x))}{\mathfrak{m}(A_{1})}=0,\qquad\forall\,\varepsilon>0.$
(4.38)

We deduce that $C_{k-\varepsilon}(x)=0$ for all $\varepsilon>0$ (cf.
Definition 4.17), and hence
$\dim_{\mathrm{geo}}(x)\geq\dim_{\mathrm{Haus}}(S_{0})$. In particular, for
any Borel subset $S_{0}\subseteq S$ with $\mathfrak{m}(S_{0})<+\infty$ it
holds

$\dim_{\mathrm{geo}}(S)\geq\dim_{\mathrm{Haus}}(S_{0}).$ (4.39)

Next, by $\sigma$-finiteness, $X=\cup_{j\in\mathbb{N}}B_{j}$ for a countable
family of Borel sets $(B_{j})_{j\in\mathbb{N}}$ with
$\mathfrak{m}(B_{j})<+\infty$. Hence, setting $S_{j}:=S\cap B_{j}$, we have
$\dim_{\mathrm{geo}}(S)\geq\dim_{\mathrm{Haus}}(S_{j})$ for all
$j\in\mathbb{N}$. Finally, we obtain

$\dim_{\mathrm{Haus}}(S)=\dim_{\mathrm{Haus}}\left(\bigcup_{j\in\mathbb{N}}S_{j}\right)=\sup_{j\in\mathbb{N}}\dim_{\mathrm{Haus}}(S_{j})\leq\dim_{\mathrm{geo}}(S),$
(4.40)

concluding the proof. ∎

The next result establishes a (sharp) geodesic dimension estimate. We will
assume the following _non-triviality condition_ :

$\mathfrak{m}(\\{z\in X\mid z\neq
x,\,\mathsf{G}(x,z)<+\infty\\})>0,\qquad\forall\,x\in\operatorname{\mathrm{supp}}\mathfrak{m}.$
(4.41)

In other words, $\operatorname{\mathrm{supp}}\mathfrak{m}$ is not a singleton
and $\mathsf{G}(x,\cdot)$ does not take the value $+\infty$ almost everywhere,
for all $x\in\operatorname{\mathrm{supp}}\mathfrak{m}$. Notice that (4.41) is
implied by the $\mathsf{MCP}(\beta)$ provided that
$\beta_{t}(+\infty)=+\infty$, see Remark 4.10.

######  Theorem 4.19 (Geodesic dimension estimate).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. with gauge function $\mathsf{G}$
satisfying the non-triviality condition (4.41). Assume that
$(X,\mathsf{d},\mathfrak{m},\mathsf{G})$ satisfies the $\mathsf{MCP}(\beta)$
condition, with $\beta$ as in (3.14). Recall that $N\in[1,+\infty)$ is the
parameter given in (3.12). Then the geodesic dimension of
$(X,\mathsf{d},\mathfrak{m})$ satisfies

$\dim_{\mathrm{geo}}(x)\leq
N,\qquad\forall\,x\in\operatorname{\mathrm{supp}}\mathfrak{m}.$ (4.42)

In particular it holds for the Hausdorff dimension:

$\dim_{\mathrm{Haus}}(\operatorname{\mathrm{supp}}\mathfrak{m})\leq\dim_{\mathrm{geo}}(\operatorname{\mathrm{supp}}\mathfrak{m})\leq
N.$ (4.43)

###### Proof.

Thanks to Lemma 4.18, estimate (4.42) implies (4.43), so it is sufficient to
prove the former. We use the half Brunn-Minkowski inequality (Theorem 4.2)
implied by the $\mathsf{MCP}(\beta)$. Fix
$x\in\operatorname{\mathrm{supp}}\mathfrak{m}$. Using that $\mathfrak{m}$ is
locally finite and the non triviality (4.41) of the gauge function, it is
immediate to see there exists a bounded Borel set $A_{1}\subset X$, with
$\mathfrak{m}(A_{1})\in(0,+\infty)$, and disjoint from $x$, satisfying

$\sup_{y\in A_{1}}\mathsf{G}(x,y)<\infty.$ (4.44)

By the half Brunn-Minkowski inequality (Theorem 4.2), it holds

$\frac{\bar{\mathfrak{m}}(A_{t})}{\mathfrak{m}(A_{1})}\geq\mathfrak{m}\shortminus\operatorname*{ess\,inf}_{y\in
A_{1}}\,\beta_{t}(\mathsf{G}(x,y)),\qquad\forall\,t\in(0,1).$ (4.45)

Let $(\theta_{n})_{n\in\mathbb{N}}$ with $\theta_{n}\in[0,+\infty)$ be such
that

$\lim_{n\to\infty}\beta_{t}(\theta_{n})=\mathfrak{m}\shortminus\operatorname*{ess\,inf}_{y\in
A_{1}}\,\beta_{t}(\mathsf{G}(x,y)).$ (4.46)

Assume first that $\theta_{n}$ is definitely bounded away from zero, so that
we can extract a subsequence such that
$\lim_{n\to\infty}\theta_{n}=\theta_{\infty}\in(0,+\infty)$, where the
finiteness of $\theta_{\infty}$ follows from (4.44). Since $\beta_{t}$ is
continuous in this range (cf. Proposition 3.4), the essential infimum above is
achieved at $\beta_{t}(\theta_{\infty})$. If
$\theta_{\infty}\in(0,\mathcal{D})$ then, by the properties (3.12) and (3.14),
we infer that

$\limsup_{t\to
0^{+}}\frac{1}{t^{N+\varepsilon}}\frac{\bar{\mathfrak{m}}(A_{t})}{\mathfrak{m}(A_{1})}\geq\limsup_{t\to
0^{+}}\frac{1}{t^{N+\varepsilon}}\beta_{t}(\theta_{\infty})=\limsup_{t\to
0^{+}}\frac{1}{t^{N+\varepsilon}}\frac{\mathsf{s}(t\theta_{\infty})}{\mathsf{s}(\theta_{\infty})}=+\infty.$
(4.47)

If $\theta_{\infty}\in[\mathcal{D},+\infty)$, then
$\beta_{t}(\theta_{\infty})=+\infty$ by (3.14), so we have a similar
conclusion.

The remaining case is when $(\theta_{n})_{n\in\mathbb{N}}$ admits a
subsequence converging to zero. So we assume that
$\lim_{n\to\infty}\theta_{n}=0$. In this case, by the properties (3.12) and
(3.14), we have

$\frac{1}{t^{N+\varepsilon}}\frac{\bar{\mathfrak{m}}(A_{t})}{\mathfrak{m}(A_{1})}\geq\lim_{n\to\infty}\frac{1}{t^{N+\varepsilon}}\beta_{t}(\theta_{n})=\lim_{n\to\infty}\frac{1}{t^{N+\varepsilon}}\frac{\mathsf{s}(t\theta_{n})}{\mathsf{s}(\theta_{n})}=\lim_{n\to\infty}\frac{(t\theta_{n})^{N}+o(t\theta_{n})^{N}}{t^{N+\varepsilon}\left(\theta_{n}^{N}+o(\theta_{n})^{N}\right)}=\frac{1}{t^{\varepsilon}}.$
(4.48)

Taking the $\limsup$ for $t\to 0^{+}$ gives $+\infty$. Combining all the
cases, we deduce that $C_{N+\varepsilon}(x)=+\infty$ for all $\varepsilon>0$
(cf. Definition 4.17), that implies (4.42). ∎

######  Remark 4.20.

The Heisenberg group $\mathbb{H}^{d}$, equipped with its Carnot-Carathéodory
distance and a Haar measure, is an example where the geodesic dimension is
strictly smaller than the Hausdorff dimension (the former being equal to
$2d+3$, while the latter is equal to $2d+2$, while the topological dimension
is $2d+1$). This is true, more in general, for all sub-Riemannian structures
that are not Riemannian. Furthermore, when equipped with a suitable gauge
function (see Section 3.3.2), the estimate provided by Theorem 4.19 is
attained.

###  4.6 Generalized Bishop-Gromov inequality

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space with gauge
function $\mathsf{G}$. Observe that the sublevel sets of a gauge function
$\mathsf{G}(x_{0},\cdot)\leq r$ may be unbounded. This occurs with the natural
gauge in the Heisenberg group (here, even $\\{\mathsf{G}(x_{0},\cdot)=0\\}$ is
unbounded). For this reason we work with the intersection with standard balls.

######  Definition 4.21.

For a point $x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ and for
$r,\rho\geq 0$, we set:

$\displaystyle\mathrm{v}_{\mathsf{G}}(x_{0},r,\rho)$
$\displaystyle:=\mathfrak{m}\left(\\{x\in X\mid\mathsf{G}(x_{0},x)\leq
r,\;\mathsf{d}(x_{0},x)\leq\rho\\}\right),$ (4.49)
$\displaystyle\mathrm{s}_{\mathsf{G}}(x_{0},r,\rho)$
$\displaystyle:=\limsup_{\delta\downarrow
0}\frac{1}{\delta}\mathfrak{m}\left(\\{x\mid\mathsf{G}(x_{0},x)\in(r-\delta,r],\;\mathsf{d}(x_{0},x)\leq\rho\\}\right).$
(4.50)

Consider also the following measure of “gauge balls” and “gauge spheres”:

$\displaystyle\mathrm{v}_{\mathsf{G}}(x_{0},r)$
$\displaystyle:=\mathfrak{m}\left(\\{x\mid\mathsf{G}(x_{0},x)\leq
r,\;\mathsf{d}(x_{0},x)\leq r\\}\right),$ (4.51)
$\displaystyle\mathrm{s}_{\mathsf{G}}(x_{0},r)$
$\displaystyle:=\limsup_{\delta\downarrow
0}\frac{1}{\delta}\left[\mathrm{v}_{\mathsf{G}}(x_{0},r)-\mathrm{v}_{\mathsf{G}}(x_{0},r-\delta)\right].$
(4.52)

Notice that
$\mathrm{v}_{\mathsf{G}}(x_{0},r)=\mathrm{v}_{\mathsf{G}}(x_{0},r,r)$ but
$\mathrm{s}_{\mathsf{G}}(x_{0},r)\neq\mathrm{s}_{\mathsf{G}}(x_{0},r,r)$.

In Figure 1 we sketch these sets for a natural gauge function in the
Heisenberg group.

Figure 1: Typical gauge sets in the Heisenberg group, for the natural choice
of gauge function described in Section 3.3.2.

######  Definition 4.22 (Meek).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. The gauge function
$\mathsf{G}:X\times X\to[0,+\infty]$ is _meek_ if for all
$\mu_{1}\in\mathcal{P}_{bs}^{*}(X,\mathsf{d},\mathfrak{m})$,
$\bar{x}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ with
$\bar{x}\notin\operatorname{\mathrm{supp}}\,\mu_{1}$, and any optimal
dynamical plan $\nu\in{\rm OptGeo}(\delta_{\bar{x}},\mu_{1})$, there exists a
Borel set $\Gamma\subset\mathrm{Geo}(X)$ with $\nu(\Gamma)=1$ such that it
holds:

$\mathsf{G}(\gamma_{0},\gamma_{t})=t\,\mathsf{G}(\gamma_{0},\gamma_{1}),\qquad\forall\,t\in(0,1],\quad\forall\,\gamma\in\Gamma.$
(4.53)

######  Remark 4.23.

The meek property is used in the proof of Theorem 4.24 below for gauge m.m.s.
satisfying the $\mathsf{MCP}(\beta)$. There, it can be replaced by a weaker
assumption where condition (4.53) is required to hold for one $\nu\in{\rm
OptGeo}(\delta_{\bar{x}},\mu_{1})$ for which the $\mathsf{MCP}(\beta)$
inequality (3.16) holds, instead than for _every_ $\nu$. This variant is
stable under the convergence of gauge metric measure spaces studied in Section
5, see proof of item (iii) of Theorem 5.17. We refrain from stating Definition
4.22 in this weaker sense since it would make it dependent on the
$\mathsf{MCP}(\beta)$ condition.

The meek property is satisfied by a large class of natural gauge functions in
sub-Riemannian geometry, see Section 7.3.1 (in particular, Theorem 7.16 and
Remark 7.17)

######  Theorem 4.24 (Generalized Bishop-Gromov).

Let $(X,\mathsf{d},\mathfrak{m})$ be a m.m.s. endowed with a gauge function
$\mathsf{G}$ satisfying the $\mathsf{MCP}(\beta)$ condition, with $\beta$ as
in (3.14). Assume that

  * •

$\operatorname{\mathrm{supp}}\mathfrak{m}$ is not a singleton;

  * •

$\mathsf{G}$ is meek; (see Remark 4.23 for a slightly weaker condition)

  * •

for all $x\in\operatorname{\mathrm{supp}}\mathfrak{m}$ and all $r>0$ it holds

$\mathfrak{m}(\\{z\in X\mid\mathsf{d}(x,z)=r,\,\mathsf{G}(x,z)=+\infty\\})=0.$
(4.54)

Then for every $x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ the
following properties hold:

  1. (i)

$\mathfrak{m}(\\{x\mid\mathsf{d}(x_{0},x)=r\\})=0$, for every $r>0$;

  2. (ii)

$\mathfrak{m}(\\{x_{0}\\})=0$; in particular, the half-Brunn-Minkowski
inequality (4.7) holds for every
$\bar{x}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ and Borel set $A\subset
X$ with $\mathfrak{m}(A)>0$;

  3. (iii)

$\mathfrak{m}(\\{x\mid\mathsf{G}(x_{0},x)=r\\})=0$, for every $r>0$;

  4. (iv)

For every $\rho>0$, the functions

$(0,\mathcal{D})\ni
r\mapsto\frac{\mathrm{s}_{\mathsf{G}}(x_{0},r,\rho)}{\mathsf{s}(r)/r}\qquad\text{and}\\\
(0,\mathcal{D}]\cap(0,+\infty)\ni
r\mapsto\frac{\mathrm{v}_{\mathsf{G}}(x_{0},r,\rho)-\mathrm{v}_{\mathsf{G}}(x_{0},0,\rho)}{\int_{0}^{r}(\mathsf{s}(t)/t)\,\mathrm{d}t},$
(4.55)

are monotone non-increasing. Here $\mathsf{s}$ is the function defining
$\beta$ as in (3.14);

  5. (v)

Assume that $\theta\mapsto\beta_{t}(\theta)$ is monotone non-increasing for
all $t\in[0,1]$, or that for all
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ it holds
$\mathsf{G}(x_{0},\cdot)\geq\mathsf{d}(x_{0},\cdot)$ $\mathfrak{m}$-a.e.. Then
the functions

$(0,\mathcal{D})\ni
r\mapsto\frac{\mathrm{s}_{\mathsf{G}}(x_{0},r)}{\mathsf{s}(r)/r}\qquad\text{and}\qquad(0,\mathcal{D}]\cap(0,+\infty)\ni
r\mapsto\frac{\mathrm{v}_{\mathsf{G}}(x_{0},r)}{\int_{0}^{r}(\mathsf{s}(t)/t)\,\mathrm{d}t}$
(4.56)

are monotone non-increasing.

  6. (vi)

For every $R_{0}>0$ there exists a constant

$\frac{1}{C_{r/R,0}}:=\inf\\{\beta_{r/R}(\theta)\mid\theta\in[0,R_{0}]\\}\in(0,(r/R)^{N}],$
(4.57)

such that it holds

$\mathrm{s}_{\mathsf{G}}(x_{0},R)\leq\frac{r}{R}C_{r/R,0}\cdot\mathrm{s}_{\mathsf{G}}(x_{0},r),\qquad\forall\,0<r\leq
R\leq R_{0}.$ (4.58)

As a consequence, the following doubling estimates for gauge balls holds:
there exists a constant $C_{0}=C_{1/2,0}>0$ such that

$\mathrm{v}_{\mathsf{G}}(x_{0},2r)\leq
C_{0}\cdot\mathrm{v}_{\mathsf{G}}(x_{0},r),\qquad\forall\,r\in[0,R_{0}/2].$
(4.59)

######  Remark 4.25.

Condition (4.54) is a mild strengthening of the non-triviality condition
(4.41). It follows from the $\mathsf{MCP}(\beta)$, provided that
$\beta_{t}(+\infty)=+\infty$, see Remark 4.10.

######  Remark 4.26.

Item (iv) can be seen as a parametric version of the Bishop-Gromov inequality
(the parameter being the “scale” $\rho$). The Bishop-Gromov estimate for
“gauge balls” and “spheres” as in item (v) holds under some additional
assumptions.

######  Remark 4.27.

We are not aware of any situation in which
$\mathrm{v}_{\mathsf{G}}(x_{0},0,\rho)$ does not vanish, but it does not seem
to be a consequence of the $\mathsf{MCP}$ condition. Thus it seems that the
extra term in item (iv) cannot be in general removed.

###### Proof.

Fix $x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ throughout the proof.

Proof of (i). Let $\delta,\rho>0$, and $0<r-\delta r<r<R-\delta R<R<+\infty$.
Let

$E:=\\{x\mid\mathsf{G}(x_{0},x)\leq\rho,\;\mathsf{d}(x_{0},x)\in(R-R\delta,R]\\},$
(4.60)

and note that $E$ is Borel. Assume $\mathfrak{m}(E)>0$. Let
$\mu_{0}=\delta_{x_{0}}$ and
$\mu_{1}=\mathfrak{m}(E)^{-1}\mathfrak{m}\llcorner_{E}$. Let $\nu\in{\rm
OptGeo}(\mu_{0},\mu_{1})$ be such that the $W_{2}$-geodesic $\mu_{t}=({\rm
e}_{t})_{\sharp}\nu$ satisfies the $\mathsf{MCP}(\beta)$ inequality. Let
$\Gamma\subseteq\mathrm{Geo}(X)$ be the corresponding set of geodesics in the
meek assumption, for which $\nu(\Gamma)=1$. Observe that
$\operatorname{\mathrm{supp}}\nu\subseteq\Gamma_{E}:={\rm
e}_{1}^{-1}(E)\cap{\rm e}_{0}^{-1}(x_{0})$ so that, letting
$\Gamma^{\prime}=\Gamma\cap\Gamma_{E}$, it holds that $\nu(\Gamma^{\prime})=1$
and the measure $\mu_{t}$ is concentrated on ${\rm e}_{t}(\Gamma^{\prime})$.
Since $\Gamma^{\prime}$ is Borel, ${\rm e}_{t}(\Gamma^{\prime})$ is Suslin.
Then, by Theorem 4.2 and Remark 4.4, we have

$\displaystyle\bar{\mathfrak{m}}({\rm
e}_{r/R}(\Gamma^{\prime}))\geq\beta_{r/R}(x_{0},E)\cdot\mathfrak{m}(E)\geq
C_{r/R,\rho}\cdot\mathfrak{m}(E),$ (4.61)

where we set

$C_{r/R,\rho}:=\inf\\{\beta_{r/R}(\theta)\mid\theta\in[0,\rho]\\}>0.$ (4.62)

The same inequality holds true trivially also if $\mathfrak{m}(E)=0$. An
important remark is that any $\nu\in{\rm OptGeo}(\mu_{0},\mu_{1})$ is
supported on a set of non-trivial geodesics, since by construction
$x_{0}\notin E$ and $({\rm e}_{0},{\rm
e}_{1})_{\sharp}\nu=\delta_{x_{0}}\otimes\mu_{1}$, so that all geodesics in
$\Gamma^{\prime}$ are non-trivial. Then, by the meek assumption, for all
$\gamma\in\Gamma^{\prime}$ it holds

$\displaystyle\mathsf{G}(x_{0},\gamma_{r/R})$
$\displaystyle=\frac{r}{R}\mathsf{G}(\gamma_{0},\gamma_{1})\in[0,r\rho/R],$
(4.63) $\displaystyle\mathsf{d}(x_{0},\gamma_{r/R})$
$\displaystyle=\frac{r}{R}\mathsf{d}(\gamma_{0},\gamma_{1})\in(r-\delta r,r].$
(4.64)

In other words, we have

$\bar{\mathfrak{m}}\left({\rm
e}_{r/R}(\Gamma^{\prime})\setminus\\{x\mid\mathsf{G}(x_{0},x)\leq
r\rho/R,\;\mathsf{d}(x_{0},x)\in(r-\delta r,r]\\}\right)=0.$ (4.65)

Since $\rho>r\rho/R$, for the given range of parameters, we have:

$\mathfrak{m}\left(\\{x\mid\mathsf{G}(x_{0},x)\leq\rho,\;\mathsf{d}(x_{0},x)\in(r-\delta
r,r]\\}\right)\geq\\\
C_{r/R,\rho}\cdot\mathfrak{m}\left(\\{x\mid\mathsf{G}(x_{0},x)\leq\rho,\;\mathsf{d}(x_{0},x)\in(R-\delta
R,R]\\}\right),$ (4.66)

where we noted that the set in the left hand side of (4.66) is
$\mathfrak{m}$-measurable. Thus for fixed $\rho\in(0,+\infty)$ the function

$(0,+\infty)\ni
r\mapsto\mathrm{v}_{\mathsf{G}}(x_{0},\rho,r)=\mathfrak{m}\left(\\{x\mid\mathsf{G}(x_{0},x)\leq\rho,\;\mathsf{d}(x_{0},x)\leq
r\\}\right),$ (4.67)

by (4.66) satisfies

$\mathrm{v}_{\mathsf{G}}(x_{0},\rho,r)-\mathrm{v}_{\mathsf{G}}(x_{0},\rho,r-\delta
r)\geq
C_{r/R,\rho}\cdot\left[\mathrm{v}_{\mathsf{G}}(x_{0},\rho,R)-\mathrm{v}_{\mathsf{G}}(x_{0},\rho,R-\delta
R)\right].$ (4.68)

Recall that, by standing assumption, $\mathfrak{m}$ is finite on bounded sets,
thus

$\mathrm{v}_{\mathsf{G}}(x_{0},\rho,r)<+\infty,\quad\text{ for any
$r,\rho>0$.}$ (4.69)

Furthermore, for fixed $\rho>0$, the function (4.67) is right-continuous, non-
decreasing, with at most countably many discontinuities. In particular there
exists arbitrarily small $r_{0}>0$ for which
$\mathrm{v}_{\mathsf{G}}(x_{0},\rho,\cdot)$ is continuous at $r_{0}$. Sending
$\delta\downarrow 0$ in (4.66), we obtain that
$\mathrm{v}_{\mathsf{G}}(x_{0},\rho,\cdot)$ is also left-continuous on
$(0,+\infty)$ and thus continuous. In particular,

$\mathfrak{m}\\{x\mid\mathsf{G}(x_{0},x)\leq\rho,\;\mathsf{d}(x_{0},x)=r\\}=0,\qquad\text{for
any $r,\rho>0$}.$ (4.70)

It follows that for any $r>0$

$\mathfrak{m}\left(\\{x\mid\mathsf{d}(x_{0},x)=r\\}\right)=\mathfrak{m}\left(\bigcup_{n\in\mathbb{N}}\\{x\mid\mathsf{G}(x_{0},x)\leq
n,\;\mathsf{d}(x_{0},x)=r\\}\right)\\\
+\mathfrak{m}\left(\\{x\mid\mathsf{G}(x_{0},x)=\infty,\;\mathsf{d}(x_{0},x)=r\\}\right)=0.$
(4.71)

The first term vanishes by (4.70), while the second one vanishes by (4.54).

Proof of (ii). By the previous item, and since
$\operatorname{\mathrm{supp}}\mathfrak{m}$ is not a singleton, any
$x_{0}\in\operatorname{\mathrm{supp}}\mathfrak{m}$ is on a sphere of positive