# Stability in the Banach isometric conjecture and nearly monochromatic
Finsler surfaces

Gautam Aishwarya<EMAIL_ADDRESS>Faculty of Mathematics, Technion -
Israel Institute of Technology, Haifa 3200003, Israel  and  Dmitry Faifman
<EMAIL_ADDRESS>School of Mathematical Sciences, Tel Aviv University,
Tel Aviv 6997801, Israel

###### Abstract.

The Banach isometric conjecture asserts that a normed space with all of its
$k$-dimensional subspaces isometric, where $k\geq 2$, is necessarily
Euclidean. The first case of $k=2$ is classical, established by Auerbach,
Mazur and Ulam using an elegant topological argument. We refine their method
to arrive at a stable version of their result: if all $2$-dimensional
subspaces are almost isometric, then the space is almost Euclidean.
Furthermore, we show that a $2$-dimensional surface, which is not a torus or a
Klein bottle, equipped with a near-monochromatic Finsler metric, is
approximately Riemannian. The stability is quantified explicitly using the
Banach–Mazur distance.

MSC classification: 52A21, 46C15, 57R15. 53C60
GA was supported by ISF grant 1468/19 and NSF-BSF grant DMS-2247834. DF was
supported by ISF grant No. 1750/20.

##  1\. Introduction and results

In 1932, Banach made the following conjecture [3].

######  Conjecture 1.1.

Let $V$ be a normed space over the reals, and $2\leq k<\dim V$ an integer.
Suppose that all linear $k$-dimensional subspaces of $V$ are isometric to each
other. Then $V$ must necessarily be a Hilbert space.

A seemingly more general conjecture, known to be equivalent to Conjecture 1.1
[19], is the following.

######  Conjecture 1.2.

Let $V$ be a finite-dimensional real vector space, and $2\leq k<\dim V$. If
$K\subset V$ is a convex body, and all its $k$-dimensional sections through a
fixed interior point of $K$ are affinely equivalent, then $K$ is an ellipsoid.

Here two subsets of two respective affine spaces are _affinely equivalent_ if
one is the image of the other under an affine map.

This conjecture has been proven in various cases, starting with the work of
Auerbach–Mazur–Ulam in 1935 [2], who proved it for $k=2$. Their proof is based
on a topological obstruction - an idea later pushed further by Gromov [9] to
settle the conjecture for all even $k$, as well as for $\dim V\geq k+2$. Much
more recently, Bor–Hernandez Lamoneda–Jimenez Desantiago–Montejano Peimbert
[4] combined topological and convex-geometric ideas to prove the conjecture
for $k=4m+1$, with the possible exception of $k=133$; while for $k=3$, the
conjecture has been confirmed by Ivanov, Mamaev, and Nordskova [12]. A local
version of Conjecture 1.1 for $k=2,3$ was established in [11, 13]. Some
results were also obtained for the Banach conjecture in complex normed spaces
[9, 5].

###  1.1. Results

In this note, we initiate the study of stability in Banach’s conjecture.
Stability results in convex geometry concern themselves either with geometric
inequalities, showing near-extremizers to be close to the extremizers; or with
the geometric characterization of a certain type of convex body, showing e.g.
that a convex body approximately satisfying a property exhibitied only by
ellipsoids, must be close to an ellipsoid. This work studies the latter type
of stability. Previous stability results of this type appeared in several
works, such as Burger and Schneider [6], Groemer [8], Gruber [10] (see also
references therein).

Loosely speaking, we wish to know if, when the linear $k$-dimensional sections
of a convex body $K$ with $0$ in its interior are approximately
linearly/affinely equivalent, $K$ must be approximately an ellipsoid. To
quantify the deviation from linear/affine equivalence, we use the Banach–Mazur
distance.

######  Definition 1.3.

Let $V,W$ be isomorphic Banach spaces (with possibly non-symmetric norms). The
Banach–Mazur distance between them is defined by

$d_{BM}(V,W)=\inf\\{\|T\|\|T^{-1}\|:T\in\operatorname{GL}(V,W)\\}.$

One can identify (non-symmetric) normed spaces with their unit balls, which
are convex bodies with the origin a fixed point in their interior. The Banach-
Mazur distance yields a corresponding notion of distance on such pointed
bodies.

######  Definition 1.4.

Let $K\subset V$, $L\subset W$ be convex bodies with the origin in their
interior. The _linear Banach–Mazur distance_ is

$d^{\operatorname{Lin}}_{BM}(K,L)=\inf\\{\lambda>0:K\subset TL\subset\lambda
K,\textnormal{ for some }T\in{\operatorname{GL}}(W,V).\\}.$

When affine maps are considered, we need to allow additional freedom in the
choice of origin. Denote by $\overline{\operatorname{GL}}(W,V)$ the set of
affine maps from $W$ to $V$. A convex body in $\mathbb{R}^{n}$ is any compact
convex set with non-empty interior.

######  Definition 1.5.

Let $K\subset V$, $L\subset W$ be convex bodies.

$d_{BM}(K,L)=\inf\\{\lambda>0:K\subset TL\subset\lambda K+z,\textnormal{ for
some }T\in\overline{\operatorname{GL}}(W,V),z\in V\\}.$

Note that if the convex bodies $K,L\subset\mathbb{R}^{n}$ are the unit balls
of norms $\|\bullet\|_{K}$, resp $\|\bullet\|_{L}$ on $\mathbb{R}^{n}$, we
have $d_{BM}(K,L)\leq
d^{\operatorname{Lin}}_{BM}(K,L)=d_{BM}(\|\bullet\|_{K},\|\bullet\|_{L})$ in
general. Equality holds e.g. when both $K,L$ are centrally symmetric. Note
also that the Banach-Mazur distance is only a non-degenerate distance function
on the quotient of the space of convex bodies with non-empty interior by
$\overline{\operatorname{GL}}(\mathbb{R}^{n})$.

The Banach–Mazur distance is usually left in multiplicative form. Thus $K,L$
are affinely equivalent if and only if $d_{BM}(K,L)=1$, and $d_{BM}(K,L)\geq
1$ in general.

We can now formulate a quantitative version of the Banach conjecture. Denote
by $\mathbb{E}^{n}$ the Euclidean $n$-dimensional space, and by $B^{n}$ the
unit ball therein.

######  Conjecture 1.6.

For any $\epsilon>0$ and $n>k\geq 2$ there is $\delta=\delta(n,k,\epsilon)>0$
such that the following holds. For any $n$-dimensional normed space $V$,
possibly non-symmetric, such that $d_{BM}(E,F)<1+\delta$ for all
$k$-dimensional linear subspaces $E,F\subset V$, it must hold that
$d_{BM}(V,\mathbb{E}^{n})<1+\epsilon$.

As before, this is the linear case of the general affine conjecture.

######  Conjecture 1.7.

For any $\epsilon>0$ and $n>k\geq 2$ there is $\delta=\delta(n,k,\epsilon)>0$
such that the following holds. For any convex body $K\subset\mathbb{R}^{n}$
with $0$ in its interior, such that $d_{BM}(K\cap E,K\cap F)<1+\delta$ for all
$k$-dimensional linear subspaces $E,F\subset\mathbb{R}^{n}$, we must have
$d_{BM}(K,B^{n})<1+\epsilon$.

We prove Conjecture 1.7 for $k=2$ with an explicit stability estimate. In the
following, $c>0$ denotes various universal constants that can be computed
explicitly.

######  Theorem 1.8.

Let $K\subset\mathbb{R}^{n}$ be a convex body with $0$ in its interior. Assume
that $d_{BM}(K\cap E,K\cap F)<1+\delta$ for all linear $2$-dimensional
subspaces $E,F\subset\mathbb{R}^{n}$. Then
$d_{BM}(K,B^{n})<1+cn^{2n^{2}}\delta^{1/6}$.

In the linear setting of Conjecture 1.6 we obtain better stability.

######  Theorem 1.9.

Let $V=\mathbb{R}^{n}$ be a normed space, not necessarily symmetric. Assume
that $d_{BM}(E,F)<1+\delta$ for all $2$-dimensional subspaces $E,F\subset V$.
Then $d_{BM}(V,\mathbb{E}^{n})<1+cn^{2}\delta^{1/3}$.

In the special case of symmetric norms this can be stated as follows.

######  Corollary 1.10.

Let $K\subset\mathbb{R}^{n}$ be a centrally-symmetric convex body. Assume that
$d_{BM}(K\cap E,K\cap F)<1+\delta$ for all linear $2$-dimensional subspaces
$E,F\subset\mathbb{R}^{n}$. Then $d_{BM}(K,B^{n})<1+cn^{2}\delta^{1/3}$.

We remark that the main interest of these results lies in the regime of fixed
$n$ and $\delta\to 0$. Indeed by the Dvoretzky–Milman theorem [1, Theorem
5.13], if $n>\exp(c\delta^{-2})$ and $K$ is a centrally-symmetric convex body,
then $K$ must have a planar section $E_{0}$ such that $d_{BM}(K\cap
E_{0},B^{2})<1+\delta$; under the assumptions of Corollary 1.10, we have
$d_{BM}(K\cap E,B^{2})<1+c\delta$ for all planar sections, yielding
$d_{BM}(K,B^{n})<1+cn^{2}\delta$ as in the proof of Theorem 1.9.

We deduce these results as a corollary of the following general theorem on
fields of convex bodies. A _continuous field of convex bodies_ over a manifold
$\Sigma$ is a family of convex bodies $K_{x}\subset T_{x}\Sigma$,
$x\in\Sigma$, continuous with respect to the Hausdorff distance within each
chart.

######  Theorem 1.11.

Let $\Sigma$ be a closed surface which is not a torus or a Klein bottle. If a
continuous field of convex bodies $(K_{x})_{x\in\Sigma}$ satisfies
$d_{BM}(K_{x},K_{y})<1+\delta$ for all $x,y\in\Sigma$, then for all
$x\in\Sigma$, $d_{BM}(K_{x},B^{2})<1+c\delta^{1/3}$.

It follows that a near-monochromatic Finsler metric on $\Sigma$ is
approximately Riemannian:

######  Corollary 1.12.

Let $\Sigma$ be as in Theorem 1.11, and assume it is equipped with a $C^{0}$
Finsler structure such that $d_{BM}(T_{x}\Sigma,T_{y}\Sigma)<1+\delta$ for all
$x,y\in\Sigma$. Then $d_{BM}(T_{x}\Sigma,\mathbb{E}^{2})<1+c\delta^{1/3}$ for
all $x\in\Sigma$.

The restriction on the topology of $\Sigma$ is necessary as one can easily
construct monochromatic Finsler structures on the torus and the Klein bottle
which are not Riemannian. It will be seen from the proof that the key required
property is the non-existence of a field of tangent lines on the surface. We
conjecture that Theorem 1.11, with some stability bounds, holds in greater
generality, in particular for even dimensional spheres.

Let us remark that the power $\frac{1}{3}$ of $\delta$ established in Theorem
1.11 is likely an artifact of the proof. It would be interesting if the sharp
power is strictly less than $1$.

The main step in deducing Theorem 1.8 from Theorem 1.11 consists of proving
the following stability result, which may be of independent interest.

######  Theorem 1.13.

Let $K\subset\mathbb{R}^{n}$ be a convex body with $0\in\mathrm{int}(K)$, such
that for all $2$-dimensional linear planes $E$, $d_{BM}(K\cap
E,B^{2})<1+\epsilon$. Then $d_{BM}(K,B^{n})<1+cn^{2n^{2}}\sqrt{\epsilon}$.

###  1.2. Ideas and plan of the proof

The proof is inspired by the proof of Auerbach–Mazur–Ulam of Conjecture 1.7
for $k=2$, $\dim V=3$, which we now recall using modern language. Setting
$K_{x}=K\cap x^{\perp}$, we get a continuous field of convex bodies in the
tangent planes of $S^{2}$, $x\mapsto K_{x}\subset T_{x}S^{2}$. A convex body
in $\mathbb{R}^{2}$ is either an ellipse, or has a finite group of affine
symmetries. Fixing $x_{0}\in S^{2}$ and assuming that none of $K_{x}$ are
ellipses, the affine maps $\mathrm{Aff}(K_{x_{0}},K_{x})$ mapping $K_{x_{0}}$
to $K_{x}$ define a covering space over $S^{2}$, which must have a section,
$x\mapsto g_{x}$, as $S^{2}$ is simply connected. Fixing a line $L\subset
T_{x_{0}}S^{2}$, and letting $L_{x}$ be the linear line in $T_{x}S^{2}$
parallel to $g_{x}(L)$, we obtain a field $x\mapsto L_{x}$ of tangent lines on
$S^{2}$, which is impossible. Thus all sections $K_{x}$ are ellipses. A
separate geometric argument, showing that a convex body all of whose
$2$-dimensional sections through an interior point are ellipses must be an
ellipsoid, completes the proof.

Since the proof of Auerbach–Mazur–Ulam, as do most other known cases of the
Banach conjecture, has a topological obstruction at its heart, it is somewhat
surprising that a stability result for $k=2$ can be obtained by building on
top of the proof outlined above. We are forced to work with approximate
isometries that do not form a group, and thus we aim to distill certain coarse
features that are sufficiently stable as to allow the application of
topological methods.

The proof of Theorem 1.11 appears in section 3. We make use of an associated
Riemannian structure on $\Sigma$ given by the Binet–Legendre (inertia)
ellipsoids $\mathcal{E}(K_{x})$. We consider sets of approximate isometries
$G_{\alpha}(K_{x},K_{y})$, consisting of affine maps $g:T_{x}\Sigma\to
T_{y}\Sigma$ that preserve the Binet–Legendre ellipsoids, and map $K_{x}$ to
$K_{y}$ up to an error $\alpha$ with respect to the Hausdorff distance. We
observe that if the elements of $G_{\alpha}(K_{x})$ form a fine net in the
orthogonal group, then $K_{x}$ must be close to an ellipse. Provided this is
not the case, the key step is then to find a range of values $\alpha$ for
which $G_{\alpha}(K_{x_{0}})$, or more precisely a low-resolution version
thereof, is independent of $\alpha$. This then allows to define a covering
space consisting of approximate isometries in
$G_{\alpha^{\prime}}(K_{x_{0}},K_{x})$, and proceed as before.

$G^{+}_{\alpha}(K_{x_{0}},K_{x})$$G^{+}_{\alpha}(K_{x_{0}},K_{y})$ Figure 1.
(Oriented) approximate isometries

The corollaries of Theorem 1.11 to stability in the Banach conjecture are
proved in section 4. To pass from Theorem 1.11 to Theorem 1.8, we prove a
stable version of independent interest of the characterization of ellipsoids
through planar sections mentioned above. For Theorem 1.9, we get better
stability by utilizing instead a stable version of the von Neumann–Jordan
theorem due to Passer.

###  1.3. Acknowledgements

The authors are indebted to Juan Carlos Alvarez Paiva for his inspiring talk
at the Integral and Metric Geometry 2022 BIRS-CMO workshop. The second-named
author is grateful to Vitali D. Milman for introducing him to the Banach
conjecture, and to Károly J. Böröczky for a helpful discussion.

##  2\. Preliminaries

By a convex body in $\mathbb{R}^{n}$ we understand a compact convex set with
non-empty interior. For an ellipsoid $\mathcal{E}$, we denote by
$d_{\mathcal{E}}$ the Hausdorff distance with respect to the Euclidean metric
for which $\mathcal{E}$ is a unit ball. Denote by $\operatorname{cent}(K)$ the
centroid (center of mass) of $K$. We say $K$ is _centered_ if
$\operatorname{cent}(K)=0$. A basic property of centered convex bodies in
$\mathbb{R}^{n}$, which we will use repeatedly, is that $-K\subset nK$ [21,
Lemma 2.3.3]. For a centered convex body $K$, the _Binet–Legendre ellipsoid_
$\mathcal{E}(K)\subset\mathbb{R}^{n}$ of $K\subset\mathbb{R}^{n}$, also called
the elipsoid of inertia, is the dual ellipsoid of the unit ball in
$(\mathbb{R}^{n})^{*}$ of the inner product

$\langle\phi,\psi\rangle_{K}=\frac{n+2}{\operatorname{vol}(K)}\int_{K}\phi(x)\psi(x)d\operatorname{vol}(x),$

where $\operatorname{vol}$ is an arbitrary Lebesgue measure on $V$. For
general $K$, we set
$\mathcal{E}(K):=\mathcal{E}(K-\operatorname{cent}(K))+\operatorname{cent}(K)$.

Evidently, if $g\in\overline{\operatorname{GL}}(\mathbb{R}^{n})$ then
$\mathcal{E}(gK)=g\mathcal{E}(K)$. If $K$ is an ellipsoid, $\mathcal{E}(K)=K$.
Note that unlike the definition in [16], which uses a fixed origin, we use the
centroid of $K$ as the center of the Binet–Legendre ellipsoid.

######  Lemma 2.1.

Let $K,L\subset\mathbb{R}^{n}$ be centered convex bodies. If
$\frac{1}{\lambda_{1}}K\subset L\subset\lambda_{2}K$, then

$\mathcal{E}(K)\subset\lambda_{1}^{\frac{n+2}{2}}\lambda_{2}^{\frac{n}{2}}\mathcal{E}(L),\quad\mathcal{E}(L)\subset\lambda_{1}^{\frac{n}{2}}\lambda_{2}^{\frac{n+2}{2}}\mathcal{E}(K).$

Furthermore, there are positive constants $a_{n},b_{n}$ such that
$a_{n}\mathcal{E}(K)\subset K\subset b_{n}\mathcal{E}(K)$. One can take
explicitly $a_{n}=2^{-\frac{n}{2}-1}n^{-\frac{3n}{4}-\frac{3}{2}}$,
$b_{n}=2^{\frac{n}{2}+1}n^{\frac{3n}{4}+\frac{3}{2}}$.

###### Proof.

We follow [17]. We have

$\int_{L}\phi(x)^{2}dx\leq\int_{\lambda_{2}K}\phi(x)^{2}dx=\lambda_{2}^{n+2}\int_{K}\phi(x)^{2}dx.$

As $\operatorname{vol}(L)\geq\lambda_{1}^{-n}\operatorname{vol}(K)$, we
conclude that

$\langle\phi,\phi\rangle_{L}\leq\lambda_{1}^{n}\lambda_{2}^{n+2}\langle\phi,\phi\rangle_{K}\Rightarrow\mathcal{E}(L)\subset\lambda_{1}^{\frac{n}{2}}\lambda_{2}^{\frac{n+2}{2}}\mathcal{E}(K).$

The second inclusion follows by symmetry.

For the second part, define $K_{s}=\frac{1}{2}(K+(-K))$. As $-K\subset nK$, we
have

$\frac{1}{n}K_{s}\subset\frac{2}{n+1}K_{s}\subset K\subset 2K_{s},$

and by the first part,

$\mathcal{E}(K_{s})\subset
2^{\frac{n}{2}}n^{\frac{n}{2}+1}\mathcal{E}(K),\quad\mathcal{E}(K)\subset
2^{\frac{n}{2}+1}n^{\frac{n}{2}}\mathcal{E}(K_{s}).$

We make use of John’s maximal volume ellipsoid $J$ of $K_{s}$, which satisfies
$J\subset K_{s}\subset\sqrt{n}J$ by John’s theorem. Observe that
$\mathcal{E}(J)=J$. By the first part, it holds that

$J\subset n^{\frac{n}{4}}\mathcal{E}(K_{s}),\quad\mathcal{E}(K_{s})\subset
n^{\frac{n+2}{4}}J.$

As

$\frac{1}{n}J\subset\frac{1}{n}K_{s}\subset K\subset 2K_{s}\subset
2\sqrt{n}J,$

we conclude that

$K\subset
2^{\frac{n}{2}+1}n^{\frac{3n}{4}+\frac{3}{2}}\mathcal{E}(K),\quad\mathcal{E}(K)\subset
2^{\frac{n}{2}+1}n^{\frac{3n}{4}+\frac{3}{2}}K.$

∎

######  Definition 2.2.

The Binet–Legendre distance between $K,L\subset\mathbb{R}^{n}$ is

$d_{BL}(K,L)=\min\\{d_{\mathcal{E}(g_{1}K)}(g_{1}K,g_{2}L):g_{1},g_{2}\in\overline{\operatorname{GL}}(\mathbb{R}^{n}),\mathcal{E}(g_{1}K)=\mathcal{E}(g_{2}L)\\}.$

######  Lemma 2.3.

$d_{BM}$ and $d_{BL}$ are equivalent distances on
$\mathcal{K}(\mathbb{R}^{n})$. Explicitly, there is $\epsilon_{n}>0$ such that
for $d_{BM}(K,L)\leq 1+\epsilon_{n}$ we have

$a_{n}(d_{BM}(K,L)-1)\leq d_{BL}(K,L)\leq\widehat{b}_{n}(d_{BM}(K,L)-1),$

where $\widehat{b}_{n}=(3n+1)(1+5n\frac{b_{n}}{a_{n}})b_{n}$.

Furthermore, for $\epsilon<\epsilon_{n}$, if $d_{BM}(K,L)=1+\epsilon$ and
$\mathcal{E}(K)=\mathcal{E}(L)=B^{n}$, then there is
$T\in\operatorname{O}(\mathbb{R}^{n})$ such that

$(1+d_{n}\epsilon)^{-1}K\subset TL\subset(1+d_{n}\epsilon)K,$

where $d_{n}=4n\frac{b_{n}}{a_{n}}(1+5n\frac{b_{n}}{a_{n}})$.

Later we will use the larger value $d_{n}=c4^{n}n^{3n+8}$ for simplicity.

###### Proof.

Write $a=a_{n},b=b_{n}$. Assume $d_{BL}(K,L)=\epsilon$. We may assume that a
Euclidean structure is fixed such that $\mathcal{E}(K)=\mathcal{E}(L)=B^{n}$,
and $d_{B^{n}}(K,L)=\epsilon$. That is, $K\subset L+\epsilon B^{n}$, $L\subset
K+\epsilon B^{n}$. Then by Lemma 2.1, $K\subset
L+\frac{\epsilon}{a}L=(1+\frac{\epsilon}{a})L$, and similarly
$L\subset(1+\frac{\epsilon}{a})K$. Thus $d_{BM}(K,L)\leq
1+\frac{\epsilon}{a}=1+\frac{d_{BL}(K,L)}{a}$.

In the other direction, assume $d_{BM}(K,L)=1+\epsilon$. We may assume
$\mathcal{E}(K)=B^{n}$, and $K\subset L\subset(1+\epsilon)K+z$ for some
$z\in\mathbb{R}^{n}$. It holds that

$q:=\operatorname{cent}(L)=\frac{1}{\operatorname{vol}(L)}\int_{L}xd\operatorname{vol}(x)=\frac{1}{\operatorname{vol}(L)}\int_{L\setminus
K}xd\operatorname{vol}(x).$

Since $K\subset(1+\epsilon)K+z$, we deduce that

$-z\in\epsilon K\Rightarrow z\in n\epsilon K.$

In particular, $|z|\leq nb\epsilon$, and
$\max_{L}|x|\leq|z|+(1+\epsilon)\max_{K}|x|\leq b(1+(n+1)\epsilon)$.

Consequently by Lemma 2.1, and since for $\epsilon<\frac{1}{n}$

$(1+\epsilon)^{n}-1=\sum_{j=1}^{n}\frac{n(n-1)\cdots(n-j+1)}{j!}\epsilon^{j}\leq\sum_{j=1}^{\infty}\frac{(n\epsilon)^{j}}{j!}=e^{n\epsilon}-1<2n\epsilon,$

it follows that for $\epsilon<\frac{1}{n+1}$,

$|q|\leq\frac{\operatorname{vol}(L\setminus
K)}{\operatorname{vol}(L)}\max_{L\setminus
K}|x|\leq\frac{((1+\epsilon)^{n}-1)\operatorname{vol}(K)}{\operatorname{vol}(K)}\max_{L}|x|\leq
4nb\epsilon$

and so $\pm q\in\frac{4nb}{a}\epsilon aB^{n}\subset\frac{4nb}{a}\epsilon K$.
Thus

$(1-4\frac{nb}{a}\epsilon)K\subset
L-q\subset(1+\epsilon)K+\frac{nb\epsilon}{a}aB^{n}+\frac{4nb\epsilon}{a}aB^{n}\subset\left(1+(1+\frac{5nb}{a})\epsilon\right)K.$

Denote $b_{n}^{\prime}=1+5\frac{nb}{a}$. Then (for $\epsilon$ sufficiently
small)

$\frac{1}{1+b_{n}^{\prime}\epsilon}K\subset
L-q\subset(1+b_{n}^{\prime}\epsilon)K,$

and it follows again by Lemma 2.1 that
$(1+b_{n}^{\prime}\epsilon)^{-n-1}B^{n}\subset\mathcal{E}(L-q)\subset(1+b_{n}^{\prime}\epsilon)^{n+1}B^{n}$,
and so we can choose a positive definite map $T$ with
$T(\mathcal{E}(L-q))=B^{n}$ with the eigenvalues of $T$ satisfying
$(1+b_{n}^{\prime}\epsilon)^{-n-1}\leq\lambda_{1},\dots,\lambda_{n}\leq(1+b_{n}^{\prime}\epsilon)^{n+1}$.
Writing $T=\operatorname{Id}+S$, it follows that the Euclidean operator norm
$\|S\|\leq 2nb_{n}^{\prime}\epsilon$ (for $\epsilon$ small enough). We
similarly have $T^{-1}=\operatorname{Id}+S^{\prime}$ with $\|S^{\prime}\|\leq
2nb_{n}^{\prime}\epsilon$.

We deduce that $L^{\prime}=T(L-q)$ has $\mathcal{E}(L^{\prime})=B^{n}$, and

$\displaystyle L^{\prime}$ $\displaystyle\subset
T(1+b_{n}^{\prime}\epsilon)K=K+b_{n}^{\prime}\epsilon
K+S(1+b_{n}^{\prime}\epsilon)K$ $\displaystyle\subset K+b_{n}^{\prime}\epsilon
bB^{n}+b(1+b_{n}^{\prime}\epsilon)S(B^{n})$ $\displaystyle\subset
K+(b_{n}^{\prime}\epsilon b+2nb_{n}^{\prime}\epsilon
b(1+b_{n}^{\prime}\epsilon))B^{n}.$

Similarly

$\displaystyle K$
$\displaystyle\subset(1+b_{n}^{\prime}\epsilon)T^{-1}L^{\prime}$
$\displaystyle\subset L^{\prime}+(b_{n}^{\prime}\epsilon
b+2nb_{n}^{\prime}\epsilon b(1+b_{n}^{\prime}\epsilon))B^{n}.$

that is, $d_{BL}(K,L)\leq
d_{B^{n}}(K,L^{\prime})\leq(3n+1)b_{n}^{\prime}b\epsilon$.

In particular, since

$L^{\prime}\subset(1+\frac{b}{a}(b_{n}^{\prime}\epsilon+2nb_{n}^{\prime}\epsilon(1+b_{n}^{\prime}\epsilon)))K,\quad
K\subset(1+\frac{b}{a}(b_{n}^{\prime}\epsilon+2nb_{n}^{\prime}\epsilon(1+b_{n}^{\prime}\epsilon)))L^{\prime},$

setting $d_{n}:=4nb_{n}^{\prime}\frac{b}{a}$ establishes the last claim of the
lemma. ∎

######  Corollary 2.4.

If $d_{BM}(K,B^{n})<1+\epsilon$ and $\operatorname{cent}(K)=0$ then
$(1+d_{n}\epsilon)^{-1}\mathcal{E}(K)\subset
K\subset(1+d_{n}\epsilon)\mathcal{E}(K)$, and consequently
$d_{BM}^{\operatorname{Lin}}(K,B^{n})<1+d_{n}\epsilon$.

Given an ellipse $\mathcal{E}\subset\mathbb{R}^{2}$, we equip
$\partial\mathcal{E}$ with the angular distance function $d_{\mathcal{E}}$
induced by the Euclidean structure for which it is a unit ball. Given two
ellipses $\mathcal{E}_{i}\subset E_{i}$, $i=1,2$, where $E_{i}$ are
$2$-dimensional linear spaces, denote by
$\operatorname{O}(\mathcal{E}_{1},\mathcal{E}_{2})$ the set of affine maps
$E_{1}\to E_{2}$ mapping $\mathcal{E}_{1}$ to $\mathcal{E}_{2}$. If
$E_{1}=E_{2}=E$, $\mathcal{E}_{1}=\mathcal{E}_{2}=\mathcal{E}$, we write
$\operatorname{O}(\mathcal{E})=\operatorname{O}(\mathcal{E},\mathcal{E})$, and
define $\operatorname{SO}(\mathcal{E})\subset\operatorname{O}(\mathcal{E})$ to
be the orientation-preserving subgroup. Note that
$\operatorname{SO}(\mathcal{E})$ is a circle with distance function
$d(g,h)=d_{\mathcal{E}}(gx,hx)$, which is independent of the choice of
$x\in\partial\mathcal{E}$. For general $\mathcal{E}_{1},\mathcal{E}_{2}$,
$\operatorname{O}(\mathcal{E}_{1},\mathcal{E}_{2})$ is the disjoint union of
two circles, and we define a metric on it by

$d(g_{1},g_{2})=\left\\{\begin{array}[]{cc}d_{\mathcal{E}_{1}}(g_{1}^{-1}g_{2},\operatorname{Id}),&\quad
g_{1}^{-1}g_{2}\in\operatorname{SO}(\mathcal{E}_{1}),\\\ \infty,&\quad
g_{1}^{-1}g_{2}\notin\operatorname{SO}(\mathcal{E}_{1}).\end{array}\right.$

Our main technical tool will be the subset of $\lambda$-approximate isometries
of convex bodies, defined as follows.

######  Definition 2.5.

Let $\lambda>0$, and let $E,F$ be linear spaces of equal dimension. For two
convex bodies $K\subset E$, $L\subset F$ we set
$G_{\lambda}(K,L)=\\{g\in\operatorname{O}(\mathcal{E}(K),\mathcal{E}(L)):d_{\mathcal{E}(L)}(L,gK)<\lambda\\}$.

We write
$G_{\lambda}(K):=G_{\lambda}(K,K)\subset\operatorname{O}(\mathcal{E}(K))$. We
can then also define the subset
$G^{+}_{\lambda}(K)=G_{\lambda}(K)\cap\operatorname{SO}(\mathcal{E}(K))$.

The following is immediate from definitions.

######  Lemma 2.6.

If $g\in G_{\lambda}(K_{1},K_{2})$ then $g^{-1}\in G_{\lambda}(K_{2},K_{1})$.
If $g_{j}\in G_{\lambda_{j}}(K_{j},K_{j+1})$ for $j=1,2$ then $g_{2}g_{1}\in
G_{\lambda_{1}+\lambda_{2}}(K_{1},K_{3})$.

We say that a set $A\subset S^{1}$ is an _$\epsilon$ -net_ if any closed
$\epsilon$-interval intersects $A$.

The following technical lemma is a stable version of the basic fact that a
convex body in $\mathbb{R}^{2}$ is either an ellipse, or has a finite group of
symmetries.

######  Proposition 2.7.

For all $\epsilon>0$, if $\alpha,\beta>0$ satisfy
$\frac{1}{a_{2}}\alpha+\frac{b_{2}}{a_{2}}\beta\leq\epsilon$ and
$G^{+}_{\alpha}(K)$ is a $\beta$-net in $\operatorname{SO}(\mathcal{E}(K))$
then $d_{BM}(K,B^{2})<1+\epsilon$.

We will later use

$\alpha(\epsilon)=\frac{a_{2}}{2}\epsilon,\quad\beta(\epsilon)=\frac{a_{2}}{2b_{2}}\epsilon,$
(1)

so that $\alpha=\alpha(\epsilon)$, $\beta=\beta(\epsilon)$ satisfy the
assumption of the proposition.

###### Proof.

Fix a Euclidean structure $|\bullet|$ on $\mathbb{R}^{2}$ whose unit ball is
$\mathcal{E}(K)$. Assume $p\in K$ satisfies $|p|=\max\\{|x|:x\in K\\}$. By
Lemma 2.1, $|p|\leq b_{2}$. Denoting $D=|p|\mathcal{E}(K)$, we have $K\subset
D$. On the other hand, since $G^{+}_{\alpha}(K)$ is a $\beta$-net, it holds
that

$D\subset
K+2|p|\sin(\frac{\beta}{2})\mathcal{E}(K)+\alpha\mathcal{E}(K)\subset\left(1+\frac{2b_{2}}{a_{2}}\sin\frac{\beta}{2}+\frac{1}{a_{2}}\alpha\right)K,$

so that
$d_{BM}(K,B^{2})=d_{BM}(K,D)<1+\frac{1}{a_{2}}\alpha+\frac{b_{2}}{a_{2}}\beta$.
∎

To pass from Theorem 1.11 to Theorem 1.9, we make use of the von
Neumann–Jordan constant, which we now recall.

######  Definition 2.8.

The _von Neumann–Jordan constant_ $C_{NJ}(V)$ of a normed space $V$ is the
least $M\geq 1$ such that for all non-zero $x,y\in V$,

$\frac{1}{M}\leq\frac{\|x+y\|^{2}+\|x-y\|^{2}}{2(\|x\|^{2}+\|y\|^{2})}\leq M.$

The celebrated von Neumann–Jordan theorem [14] asserts that $C_{NJ}(V)=1$
implies $V$ is Euclidean. The opposite direction is trivial, asserting simply
the validity of the parallelogram identity in Euclidean space.

We will need a stable version of these facts, comparing the von Neumann–Jordan
constant with the Banach–Mazur distance to Euclidean space. The fact that
near-Euclidean normed spaces satisfy an approximate parallelogram identity is
easy, and appeared e.g. in [15, Theorem 5].

######  Lemma 2.9.

For any normed space $V$ it holds that $C_{NJ}(V)\leq
d_{BM}(V,\mathbb{E}^{n})^{2}$.

A stable version of the von Neumann–Jordan theorem was established by Passer.

######  Theorem 2.10 (Passer [20], Theorem 1.2).

For an $n$-dimensional, symmetric normed space $V$ it holds that
$d_{BM}(V,\mathbb{E}^{n})\leq 1+cn^{2}(C_{NJ}(V)-1)$.

Here as before $c>0$ is an explicit universal constant.

##  3\. Fields of convex bodies on a surface

We now prove Theorem 1.11. It will be more convenient to work with the
following equivalent formulation.

######  Theorem 3.1.

Let $\Sigma$ be a smooth closed surface which is not a torus or a Klein
bottle. For any $\delta>0$ and any continuous field of convex bodies
$(K_{x})_{x\in\Sigma}$, $K_{x}\subset T_{x}\Sigma$, such that
$d_{BM}(K_{x},K_{y})<1+\delta$ for all $x,y\in\Sigma$, there must exist
$x\in\Sigma$ such that $d_{BM}(K_{x},B^{2})<1+c\delta^{1/3}$.

Here and in the proof, $c,C$ are explicit universal constants which assume
different values in distinct appearances, whose values we omit for simplicity.

###### Proof.

We denote by $d$ the distance function on various circles appearing in the
proof.

Let $(K_{x})$ be a field of convex bodies over $\Sigma$ such that
$d_{BM}(K_{x},K_{y})<1+\delta$ for all $x,y$, and assume that
$d_{BM}(K_{x},B^{2})\geq 1+\epsilon$, for all $x$. We may assume that
$\operatorname{cent}(K_{x})=0$ for all $x$.

Step 1: choose parameters $\alpha_{3},\beta_{0}$ such that
$G_{\alpha_{3}}(K_{x},K_{y})$ does not contain a pair of elements
$g_{1},g_{2}$ with $d(g_{1},g_{2})\in[\beta_{0}/2,\beta_{0}]$.

Denote $\alpha_{0}=\alpha(\epsilon)$, $\beta_{0}^{\prime}=\beta(\epsilon)$ as
in eq. (1). Let $\beta_{0}\leq\beta_{0}^{\prime}$ be the largest real of the
form $\beta_{0}=\frac{2\pi}{B_{0}}$ with $B_{0}\geq 3$ an integer. By
Proposition 2.7, $G^{+}_{\alpha_{0}}(K_{x})$ is not a $\beta_{0}$-net. Note
that $\alpha_{0},\beta_{0}\in[c\epsilon,C\epsilon]$.

Denote $\alpha_{1}=\alpha_{0}/(2B_{0})$, so that
$c\epsilon^{2}\leq\alpha_{1}\leq C\epsilon^{2}$. It follows that
$G^{+}_{\alpha_{1}}(K_{x})$ does not contain elements $g$ with
$d(g,\operatorname{Id})\in[\beta_{0}/2,\beta_{0}]$: if $g$ were such an
element, then the iterates $g^{j}\in G^{+}_{\alpha_{0}}(K_{x})$ for
$j=0,1,\dots,2B_{0}$ form a $\beta_{0}$-net, in contradiction.

Denote $\alpha_{2}=\alpha_{1}/2$, $\beta_{1}=\beta_{0}/2$. It follows that the
distance between $g_{1},g_{2}\in G_{\alpha_{2}}(K_{x})$ cannot lie in
$[\beta_{1},\beta_{0}]$, or else $g_{1}^{-1}g_{2}\in
G^{+}_{\alpha_{1}}(K_{x})$, while
$d(g_{1}^{-1}g_{2},\operatorname{Id})\in[\beta_{1},\beta_{0}]$, in
contradiction.

Using Lemma 2.3, we let $\delta^{\prime}=b_{2}^{\prime}\delta$ be the upper
bound on $d_{BL}(K,L)$ when $d_{BM}(K,L)\leq 1+\delta$. Assume

$\alpha_{3}:=\alpha_{2}-\delta^{\prime}>8B_{0}\delta^{\prime}\iff\delta^{\prime}<\frac{1}{8B_{0}+1}\alpha_{2},$
(2)

which holds when $\delta<C\epsilon^{3}$, as we will henceforth assume to
arrive at a contradiction. The reason for the choices in eq. (2) becomes
evident at step 2 of the proof; presently we observe that
$\alpha_{3}>\delta^{\prime}$, and so $G_{\alpha_{3}}(K_{x},K_{y})$ is non-
empty. Furthermore, $G_{\alpha_{3}}(K_{x},K_{y})$ does not contain a pair of
elements $g_{1},g_{2}$ with $d(g_{1},g_{2})\in[\beta_{1},\beta_{0}]$: fixing
$h\in\operatorname{O}(\mathcal{E}(K_{x}),\mathcal{E}(K_{y}))$ such that
$d_{BL}(K_{x},K_{y})=d_{\mathcal{E}(K_{y})}(hK_{x},K_{y})$, we would otherwise
have elements $\hat{g}_{1}:=g_{1}h^{-1},\hat{g}_{2}:=g_{2}h^{-1}\in
G_{\alpha_{3}+\delta^{\prime}}(K_{y})=G_{\alpha_{2}}(K_{y})$, with
$d(\hat{g}_{1},\hat{g}_{2})\in[\beta_{1},\beta_{0}]$, in contradiction.

Step 2: find a range of parameters $\alpha$ for a fixed point $x_{0}\in\Sigma$
for which $G_{\alpha}(K_{x_{0}})$ has the same $\beta_{1}$-clusters.

By a _$\beta$ -cluster_ in $A\subset S^{1}$ we understand a maximal subset of
$A$ of diameter at most $\beta$.

Recall that
$G_{\alpha_{3}}(K_{x},K_{y})\subset\operatorname{O}(\mathcal{E}(K_{x}),\mathcal{E}(K_{y}))$,
and the latter is the disjoint union of two circles equipped with a metric.

Let $0<\alpha\leq\alpha_{3}$. We say that $g_{1},g_{2}\in
G_{\alpha}(K_{x},K_{y})$ are $\beta_{1}$-close if $d(g_{1},g_{2})<\beta_{1}$.
By construction, this is an equivalence relation on $G_{\alpha}(K_{x},K_{y})$.
The $\beta_{1}$-close equivalence classes are precisely the
$\beta_{1}$-clusters of $G_{\alpha}(K_{x},K_{y})$, which are therefore
pairwise disjoint. Moreover, any two $\beta_{1}$-clusters in
$G_{\alpha}(K_{x},K_{y})$ are separated by a distance of at least $\beta_{0}$.

Given $g\in G_{\alpha}(K_{x},K_{y})$, we write $[g]$ for the unique
$\beta_{1}$-cluster containing $g$. Furthermore, for $\alpha\leq\alpha_{3}$
and $0<\eta<\alpha$, if $Z\subset G_{\alpha-\eta}(K_{x},K_{y})$ is a
$\beta_{1}$-cluster, and $h\in G_{\eta}(K_{y},K_{z})$, write $h[Z]:=[h\circ
g]\subset G_{\alpha}(K_{x},K_{z})$, where $g\in Z$ can be arbitrary.

The number of $\beta_{1}$-clusters in $G_{\alpha}(K_{x},K_{y})$ is a non-
decreasing function of $\alpha\in(0,\alpha_{3}]$, which is bounded from above
by $2B_{0}$. Given $0<\alpha<\alpha^{\prime}\leq\alpha_{3}$, there is a
natural injective map from the $\beta_{1}$-clusters of
$G_{\alpha}(K_{x},K_{y})$ into the $\beta_{1}$-clusters of
$G_{\alpha^{\prime}}(K_{x},K_{y})$, given by set inclusion.

Fix $x_{0}\in\Sigma$. Since $\alpha_{3}>8B_{0}\delta^{\prime}$, we can find
$\gamma_{0}\in[0,\alpha_{3}-3\delta^{\prime}]$ such that the number of
$\beta_{1}$-clusters in $G_{\alpha}(K_{x_{0}})$ is constant for
$\alpha\in[\gamma_{0},\gamma_{0}+3\delta^{\prime}]$. Putting
$\gamma_{1}=\gamma_{0}+3\delta^{\prime}$, the $\beta_{1}$-clusters of
$G_{\alpha}(K_{x_{0}})$ are therefore naturally identified across the range
$\alpha\in[\gamma_{0},\gamma_{1}]$.

Step 3: construct a covering space of $\Sigma$ out of the $\beta_{1}$-clusters
in $G_{\gamma_{0}}(K_{x_{0}})$.

For all $x\in\Sigma$, choose
$h_{x}\in\operatorname{O}(\mathcal{E}(K_{x_{0}}),\mathcal{E}(K_{x}))$ such
that
$d_{\mathcal{E}(K_{x})}(h_{x}K_{x_{0}},K_{x})<\frac{3}{2}\delta^{\prime}$. The
map $Z\mapsto h_{x}[Z]$ gives an embedding of the set of $\beta_{1}$-clusters
in $G_{\gamma_{0}}(K_{x_{0}})$ into the set of $\beta_{1}$-clusters in
$G_{\gamma_{1}}(K_{x_{0}},K_{x})$. Furthermore, the image $C_{x}$ of this
embedding is independent of the choice of $h_{x}$: if $\hat{h}_{x}$ is another
such map, and $g\in G_{\gamma_{0}}(K_{x_{0}})$, then
$g^{\prime}=\hat{h}_{x}^{-1}h_{x}g\in G_{\gamma_{1}}(K_{x_{0}})$. Recall by
construction that we may find $g^{\prime\prime}\in G_{\gamma_{0}}(K_{x_{0}})$
with $d(g^{\prime},g^{\prime\prime})<\beta_{1}$. It then follows that
$d(h_{x}g,\hat{h}_{x}g^{\prime\prime})=d(\hat{h}_{x}g^{\prime},\hat{h}_{x}g^{\prime\prime})<\beta_{1}$,
and so $\hat{h}_{x}g^{\prime\prime}$ and $h_{x}g$ define the same
$\beta_{1}$-cluster in $G_{\gamma_{1}}(K_{x_{0}},K_{x})$, proving the claim.
Let $m$ be the number of elements (clusters) in $C_{x}$, which is the same for
all $x$ by construction.

Denote by $p:\widehat{C}\to\Sigma$ the bundle with fiber $C_{x}$ over
$x\in\Sigma$. Our next goal will be topologize $\widehat{C}$ so as to make it
a covering space of $\Sigma$. We will next describe a natural identification
of the fibers of $p$ over small subsets of $\Sigma$.

Fix any Riemannian metric on $\Sigma$, and denote by $B_{r}(z)$ an open ball
of radius $r$ centered at $z$ in this metric. Take $r_{0}$ small enough so
that $B_{r_{0}}(z)$ can be identified with an open subset of $T_{z}\Sigma$
through the exponential map, for all $z\in\Sigma$. For $x,y\in B_{r_{0}}(z)$,
$x=\exp_{z}(\xi)$, $y=\exp_{z}(\eta)$, we let
$\overline{u}_{xy}^{z}:T_{z}\Sigma\to T_{z}\Sigma$ be the unique positive
definite map, with respect to the inner product determined by
$d_{\xi}\exp^{-1}(\mathcal{E}(K_{x}))$, mapping
$d_{\xi}\exp^{-1}(\mathcal{E}(K_{x}))$ to
$d_{\eta}\exp^{-1}(\mathcal{E}(K_{y}))$.

Thus $u_{xy}^{z}:=d_{\eta}\exp\circ\overline{u}_{xy}^{z}\circ
d_{\xi}\exp^{-1}\in\operatorname{SO}(\mathcal{E}(K_{x}),\mathcal{E}(K_{y}))$.
Decreasing $r_{0}$ further, we can achieve $u_{xy}^{z}\in
G^{+}_{\delta^{\prime}/4}(K_{x},K_{y})$ for all $x,y\in B_{r_{0}}(z)$ by the
continuity of $x\mapsto K_{x}$, as well as
$d(u_{xy}^{z},u_{xy}^{z^{\prime}})<\frac{\epsilon}{2}$ for $x,y\in
B_{r_{0}}(z)\cap B_{r_{0}}(z^{\prime})$.

Choose $h_{x}\in G_{\delta^{\prime}}(K_{x_{0}},K_{x})$. Now if $Z$ is a
$\beta_{1}$-cluster in $G_{\gamma_{0}}(K_{x_{0}})$, then $h_{x}[Z]$ is a
$\beta_{1}$-cluster in $C_{x}$, while $u_{xy}^{z}\circ h_{x}[Z]$ is a
$\beta_{1}$-cluster in $C_{y}$, and we define the bijection
$b_{xy}:p^{-1}(x)\to p^{-1}(y)$ by setting $b_{xy}(h_{x}[Z])=u^{z}_{xy}\circ
h_{x}[Z]$, for all $Z$. One readily checks that $b_{xy}$ is independent of the
choice of $h_{x}$, that $b_{xy}$ does not depend on $z$ as long as $x,y\in
B_{r_{0}}(z)$, and that $b_{yx^{\prime}}\circ b_{xy}=b_{xx^{\prime}}$ for all
$x,y,x^{\prime}\in B_{r_{0}}(z)$.

This now allows us to topologize $\widehat{C}$: Taking $U\subset B_{r_{0}}(z)$
open, $p^{-1}(U)$ is, by the above, the disjoint union of $m$ copies
$U_{1},\dots,U_{m}$ of $U$. The collection of all such $U_{j}$ is a basis of
the topology of $\widehat{C}$. With this topology, $p:\widehat{C}\to\Sigma$ is
an $m$-sheeted covering map. Moreover, $\widehat{C}$ naturally inherits a
smooth structure from $\Sigma$ such that $p$ is a local diffeomorphism.

The bundle $p^{*}\widehat{C}$ over $\widehat{C}$ trivially admits a section
$y\mapsto J_{y}$. Recall that $J_{y}\subset
G_{\gamma_{1}}(K_{x_{0}},K_{p(y)})$.

Step 4: find a continuous section. We next choose a continuous field of closed
intervals $\widehat{C}\ni y\mapsto
I_{y}\subset\operatorname{O}(\mathcal{E}(K_{x_{0}}),\mathcal{E}(K_{p(y)}))$ of
length $3\beta_{1}$ such that $J_{y}\subset I_{y}$. This can be done as
follows. Choose a finite cover of $\widehat{C}$ by $r_{0}$-balls
$U_{j}=B_{r_{0}}(z_{j})$. Let $\rho_{j}$ be a subordinate partition of unity.
In each $U_{j}$, one can easily choose a field of $3\beta_{1}$-intervals
$I^{j}_{y}$ for $y\in U_{j}$ such that $J_{y}\subset I^{j}_{y}$, and
$u^{z_{j}}_{p(y)p(z)}(I^{j}_{y})\subset\operatorname{O}(\mathcal{E}(K_{x_{0}}),\mathcal{E}(K_{p(z)}))$
is independent of $y$, if $r_{0}$ is sufficiently small. Then define
$I_{y}:=\sum_{j}\rho_{j}(y)I^{j}_{y}$, namely the weighted mean of the
intervals $I^{j}_{y}$ with weights $\rho_{j}(y)$, which is well defined since
all of the intervals $I^{j}_{y}$ are contained in a fixed interval
$I_{y}^{\prime}$ of length $2\beta_{1}+\beta_{1}+2\beta_{1}=5\beta_{1}<2\pi$
containing $J_{y}$, and so the averaging is carried out inside
$I^{\prime}_{y}$. Define $P_{y}$ as the center of $I_{y}$.

Thus $P_{y}$ is a continuous global section over $\widehat{C}$ of the
principal $\operatorname{O}(2)$-bundle with fiber
$\operatorname{O}(\mathcal{E}(K_{x_{0}}),\mathcal{E}(K_{p(y)}))\subset\operatorname{GL}(T_{x_{0}}\Sigma,T_{y}\widehat{C})$
over $y\in\widehat{C}$, which readily implies that $\widehat{C}$ admits a
continuous field of tangent lines: taking any tangent line $L\subset
T_{x_{0}}\Sigma$, $P_{y}L$ defines such a field. However
$\chi(\widehat{C})=m\chi(\Sigma)\neq 0$, a contradiction.

This contradiction means that one cannot have $\delta<C\epsilon^{3}$,
concluding the proof.

∎

Proof of Corollary 1.12. Denote by $K_{x}\subset T_{x}\Sigma$ the unit ball of
the Finsler structure. By Theorem 1.11, $d_{BM}(K_{x},B^{2})<1+c\delta^{1/3}$
for all $x\in\Sigma$.

The assignment $x\mapsto\operatorname{cent}(K_{x})$ defines a vector field
over $\Sigma$, which must vanish somewhere by the Poincaré-Hopf theorem since
by assumption $\chi(\Sigma)\neq 0$. Taking $z$ to be a point where
$\operatorname{cent}(K_{z})=0$, we deduce by Corollary 2.4 that
$d_{BM}^{\operatorname{Lin}}(K_{z},B^{2})<1+c\delta^{1/3}$. As
$d_{BM}^{\operatorname{Lin}}(K_{x},K_{z})<1+\delta$ for all $x$, the
conclusion follows. ∎

##  4\. The stable Banach conjecture for planar sections

Proof of Theorem 1.9. Let $K$ be the unit ball of $V$. Let $F\subset V$ be a
$3$-dimensional subspace. By Theorem 1.11, $d_{BM}(K\cap E,B^{2})\leq
1+c\delta^{1/3}$ for all planes $E\subset F$

Fix a Euclidean structure on $F$, and denote by $S(F)$ the unit sphere. For a
$2$-dimensional subspace $\theta^{\perp}\subset F$ with $\theta\in S(F)$,
define $p(\theta):=\operatorname{cent}(K\cap\theta^{\perp})\in\theta^{\perp}$.
As $p(\theta)$ defines a continuous vector field over $S(F)$, it must vanish
somewhere. Let $E_{0}\subset F$ be a plane such that
$\operatorname{cent}(K\cap E_{0})=0$.

By Corollary 2.4, there is an absolute constant $c>0$ such that

$(1+c\delta^{1/3})^{-1}\mathcal{E}(K\cap E_{0})\subset K\cap
E_{0}\subset(1+c\delta^{1/3})\mathcal{E}(K\cap E_{0}).$

In particular, $K\cap E_{0}$ is nearly centrally-symmetric:

$-(K\cap E_{0})\subset(1+c\delta^{1/3})(K\cap E_{0}).$

By assumption, it holds for any other linear plane $E\subset V$ that
$d^{\operatorname{Lin}}_{BM}(K\cap E,K\cap E_{0})<1+\delta$, and it easily
follows that $-(K\cap E)\subset(1+c\delta^{1/3})(K\cap E)$ for all $E$.
Therefore, $-K\subset(1+c\delta^{1/3})K$.

Let $K_{s}=\frac{1}{2}(K+(-K))$ be the Minkowski symmetrization of $K$. It
follows readily that

$(1+c\delta^{1/3})^{-1}K_{s}\subset K\subset(1+c\delta^{1/3})K_{s}.$ (3)

It follows by the above that $d_{BM}(K_{s}\cap E,B^{2})<1+c\delta^{1/3}$ for
all linear planes $E$. By Lemma 2.9, the von Neumann–Jordan constant of each
section $K_{s}\cap E$ satisfies $C_{NJ}(K_{s}\cap E)<1+c\delta^{1/3}$.
Consequently, $C_{NJ}(K_{s})<1+c\delta^{1/3}$. By Theorem 2.10,

$\displaystyle
d_{BM}(K_{s},B^{n})=d^{\operatorname{Lin}}_{BM}(K_{s},B^{n})<1+cn^{2}\delta^{1/3},$

and by eq. (3), $d_{BM}^{\operatorname{Lin}}(K,B^{n})<1+cn^{2}\delta^{1/3}$. ∎

To deduce Theorem 1.8, we will replace Passer’s theorem with Theorem 1.13,
which we now proceed to prove and restate here for convenience. It is a stable
version of the simple geometric fact proved in [2] (see also [7, Lemma
16.12]), asserting that if the $2$-dimensional sections of a convex body
through a fixed interior point are all ellipses, then the body is an
ellipsoid.

######  Theorem 4.1.

Let $K\subset\mathbb{R}^{n}$ be a convex body with $0\in\mathrm{int}(K)$ such
that for all $2$-dimensional linear planes $E$, $d_{BM}(K\cap
E,B^{2})<1+\epsilon$. Then $d_{BM}(K,B^{n})<1+C_{n}\sqrt{\epsilon}$, where
$C_{n}=cn^{2n^{2}}$.

###### Proof.

We prove the statement with $C^{\prime}_{n}=c^{n}d_{n-1}d_{n-2}\cdots d_{2}$
replacing $C_{n}$, as $C^{\prime}_{n}<C_{n}$. We proceed by induction on $n$,
with the base $n=2$ being trivial.

First consider the map $\operatorname{Gr}_{n-1}(\mathbb{R}^{n})\ni
H\mapsto\operatorname{cent}(K\cap H)\in H$. This is a global section of the
vector bundle $\gamma^{\perp}$ with fiber $H$ over
$\mathbb{R}\mathbb{P}^{n-1}=\mathbb{P}((\mathbb{R}^{n})^{*})$, which has full
Stiefel-Whitney class $w(\gamma^{\perp})=1+a+\dots+a^{n-1}$, where $a\in
H^{1}(\mathbb{R}\mathbb{P}^{n-1},\mathbb{Z}_{2})$ is the generator (see e.g.
[18, chapter 4]). It follows that any global section must vanish somewhere,
and we fix a hyperplane $H_{0}$ such that $\operatorname{cent}(K\cap
H_{0})=0$. Denote $\mathcal{E}_{0}=\mathcal{E}(K\cap H_{0})$.

By the induction hypothesis, $d_{BM}(K\cap
H_{0},B^{n-1})<1+C^{\prime}_{n-1}\sqrt{\epsilon}$. By Corollary 2.4,

$(1+d_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})^{-1}\mathcal{E}_{0}\subset K\cap
H_{0}\subset(1+d_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})\mathcal{E}_{0}.$ (4)

Since any chord of $\mathcal{E}_{0}$ through the origin is bisected by it, it
follows that if $[A,A^{\prime}]$ is a chord of $K\cap H_{0}$ through the
origin, then for any Euclidean norm one has

$(1+d_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})^{-1}\leq\frac{|A|}{|A^{\prime}|}\leq
1+d_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}$ (5)

Let $H_{1},H_{2}$ be the affine hyperplanes supporting $K$ and parallel to
$H_{0}$. Assume $H_{1}$ is at least as far from the origin as $H_{2}$ (with
respect to any norm on $\mathbb{R}^{n}/H_{0}$). Fix $z\in\partial K\cap
H_{1}$.

$H_{0}$$H_{1}$$H_{2}$$z$$o$$E$$L$$L_{1}$$L_{2}$$K$ Figure 2.

Now let $E$ be any linear $2$-dimensional plane through $z$, and define
$L=E\cap H_{0}$. Denote $\mathcal{E}_{E}:=\mathcal{E}(K\cap E)$,
$p_{E}:=\operatorname{cent}(\mathcal{E}_{E})=\operatorname{cent}(K\cap E)$. We
will use the Euclidean structure on $E$ with unit ball
$\mathcal{E}_{E}-p_{E}$. Note that by Corollary 2.4,

$(1+c\epsilon)^{-1}(\mathcal{E}_{E}-p_{E})\subset K\cap
E-p_{E}\subset(1+c\epsilon)(\mathcal{E}_{E}-p_{E}).$ (6)

Let $L_{i}=H_{i}\cap E$, $i=1,2$ be the lines in $E$ which are supporting
lines of $K\cap E$ and are parallel to $L$. As the distance between $L_{1}$
and $L_{2}$ is at least the diameter of
$(1+c\epsilon)^{-1}(\mathcal{E}_{E}-p_{E})$, it follows that

$\mathrm{dist}(L_{1},0)\geq 1-c\epsilon.$ (7)

We will repeatedly make use of the following simple fact: If two concentric
discs $B_{1},B_{2}$ of radii $1-c\epsilon$, resp. $1+C\epsilon$ are given, and
$L$ is a line, then for $\epsilon>0$, we have

$\mathrm{Length}(L\cap(B_{2}\setminus B_{1}))<c\sqrt{\epsilon}.$ (8)

Orient $L$ arbitrarily. Denote $[A,A^{\prime}]=K\cap L$,
$[A_{E},A^{\prime}_{E}]=\mathcal{E}_{E}\cap L$. By eq. (5),

$|\frac{A+A^{\prime}}{2}-0|\leq cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}.$

By eqs. (6) and (8), $|A-A_{E}|\leq c\sqrt{\epsilon}$, and
$|A^{\prime}-A_{E}^{\prime}|\leq c\sqrt{\epsilon}$. Consequently, setting
$0_{E}:=(A_{E}+A^{\prime}_{E})/2$, we find

$|0_{E}-0|\leq cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}.$ (9)

Let $z_{E}$ be the touching point on $\partial\mathcal{E}_{E}$ of a translate
of $L$ that is on the same side of $L$ as $L_{1}$. By eq. (8),

$|z-z_{E}|\leq c\sqrt{\epsilon}.$ (10)

Let $[z,w]$ be the chord in $K\cap E$ defined by the line through $z,0$. Let
$[z_{E},w_{E}]$ be the chord in $\mathcal{E}_{E}$ defined by the line through
$z_{E},0_{E}$. Let $p$ be the midpoint of $[z,w]$, and note that the midpoint
of $[z_{E},w_{E}]$ is $p_{E}$.

$L$$L_{1}$$L_{2}$$z$$o$$p_{E}$$\mathcal{E}_{E}$$A$$A^{\prime}$$A_{E}$$A_{E}^{{}^{\prime}}$$o_{E}$$w$$z_{E}$$w_{E}$$p$$w^{\prime}$$E$
Figure 3. Constructions in the plane $E$

We next claim that

$|w-w_{E}|\leq cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}.$ (11)

Denoting by $w^{\prime}$ the intersection of the ray $R$ from $z$ to $w$ with
$\partial\mathcal{E}_{E}$, it holds by eq. (8) that $|w-w^{\prime}|\leq
c\sqrt{\epsilon}$. Next, from eq. (7) we conclude that the angle formed by the
ray $R_{E}$ from $z_{E}$ to $0_{E}$ with $\partial\mathcal{E}_{E}$ lies in
$[\frac{\pi}{4},\frac{\pi}{2}]$. From eqs. (9), (10), and (7) we deduce that
the angle between $R$ and $R_{E}$ is at most
$cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}$. As
$\mathrm{dist}(R\cap\mathcal{E}_{E},R_{E}\cap\mathcal{E}_{E})\leq
c\sqrt{\epsilon}$, we conclude that $|w_{E}-w^{\prime}|\leq
cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}$, and so
$|w-w_{E}|\leq|w-w^{\prime}|+|w^{\prime}-w_{E}|\leq
cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}$, showing eq. (11).

We deduce from eqs. (10), (11) that also $|p-p_{E}|\leq
cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon}$.

From (6) it follows that $K\cap E$ is close to an ellipse with a center at
$p$:

$(1+cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})^{-1}(\mathcal{E}_{E}-p)\subset
K\cap E-p\subset(1+cd_{n-1}C_{n-1}\sqrt{\epsilon})(\mathcal{E}_{E}-p).$ (12)

We now let $\mathcal{E}$ be the unique ellipsoid with center at $p$ such that
both $z$ and $\partial\mathcal{E}_{0}$ lie on $\partial\mathcal{E}$. It
follows from eqs. (12), (4), (10), (7) that

$(1+cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})^{-1}(\mathcal{E}_{E}-p)\subset(\mathcal{E}-p)\cap
E\subset(1+cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})(\mathcal{E}_{E}-p),$

and thus also

$(1+cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})^{-1}(\mathcal{E}-p)\subset
K-p\subset(1+cd_{n-1}C^{\prime}_{n-1}\sqrt{\epsilon})(\mathcal{E}-p).$

Taking $C^{\prime}_{n}=cd_{n-1}C^{\prime}_{n-1}$ completes the induction and
the proof.

∎

Proof of Theorem 1.8. From Theorem 1.11 it follows that $d_{BM}(K\cap
E,B^{2})<1+c\delta^{1/3}$ for all linear planes $E$. We then deduce from
Theorem 4.1 that $d_{BM}(K,B^{n})<1+cn^{2n^{2}}\delta^{1/6}$, concluding the
proof.∎

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