Analysis and Convex Geometry Week at UniAndes
February 17-21, 2025
Bogota, Colombia
Description
The interplay between Analysis and Convex Geometry is a very active and broad subject. Connections arise with
Asymptotic Geometric Analysis, Geometric Tomography, Probability, Information Theory, Combinatorics, as well as other branches
of Mathematics, Statistics, Computer Science, and Physics.
Analytic methods in Convex Geometry are vast and fruitful. The conference will cover a variety of exciting topics
ranging from Geometric Tomography, Functional and Harmonic Analysis, to Probabilistic aspects of Convexity related to the
Concentration of Measure phenomena, Random Matrix theory and Information Theory. The plenary talks will be delivered by some
of the most prominent researchers in the field as well as a number of junior participants.
The conference will serve as a wonderful opportunity to bring together leading experts and junior researchers to exchange ideas,
showcase their work and foster further fruitful collaborations.
We hope that this conference will serve as a starting point for a local network of researchers and students and will be
followed-up in coming years by futher meetings of various types such as conferences, workshops, seminars and schools.
Main objectives
- Promote the cutting-edge research in convex geometry in Latin America.
- Introduce students and young researchers to the field and connect them with internationally renowned senior researchers.
- Foster collaborations in the region and establish a local network of collaborators and students in the field.
- Strengthen the ties between Colombia and the international mathematical community.
Organizers:
Susanna Dann (Universidad de los Andes, Bogota, Colombia)
Maria Angeles Hernandez Cifre (Universidad de Murcia, Murcia, Spain)
Artem Zvavitch (Kent State University, Kent, Ohio, USA)
Registration
Please register if you plan to attend regardless of whether you need financial support or not.
We have applied, looking for and will be applying for funds to cover travel and accommodation expenses.
Please indicate in the registration form if you need support.
We will come back to you as soon as possible with an answer.
| Registration form |
Programme
| Monday | Tuesday | Wednesday | Thursday | Friday | |
| 08:00-08:45 | registration | registration | |||
| 08:45-09:00 | welcome | registration | |||
| 09:00-9:45 | S. Bobkov | C. Buchta | I. Barany | G. Livshyts | E. Werner |
| 09:45-10:15 | coffee/discussion | coffee/discussion | coffee/discussion | coffee/discussion | coffee/discussion |
| 10:15-10:40 | P. Simanjuntak | L. Radamacher | A. Burchard | L. Brauner | M. Rapaport |
| 10:40-11:05 | M. Mouamine | F. Marín Sola | F. Moreira | A. Ilienko | Z. Lángi | 11:05-11:30 | J. Ulivelli | B. Jaye | O. Ortega | B. Zawalski | A. Stancu |
| 11:30-11:55 | lunch break | lunch break | group foto | lunch break | M. Runge |
| 11:55-14:00 | lunch break | lunch break | free afternoon/ excursion (Gold Museum, Botero Museum, Candelaria) |
lunch break | closing toast |
|---|---|---|---|---|---|
| 14:00-14:45 | G. Kur | A. Litvak | A. Eskenazis | ||
| 14:55-15:40 | E. Morales | D. Galicer | D. Faifman | ||
| 15:40-16:10 | coffee/discussion | coffee/discussion | coffee/discussion | ||
| 16:10-16:55 | G. Ivanov | G. Ambrus | R. Brandenberg | ||
| 16:55-17:50 | M. Naszodi | A. Sagmeister | C. Schütt | ||
| reception (18:00-20:00) |
conference dinner (19:00-21:00) |
Talks
Gergely Ambrus, University of Szeged and Alfréd Rényi Mathematical Institute, Hungary
Cube sections and the Laplace-Pólya integral
Abstract: Calculating and estimating the volume of hyperplane sections of the cube has been in the center of attention for over a century. Most results rely on an integral formula for the volume that dates back to Pólya,
and has its origins in Laplace's work in probability. We consider central hyperplane sections and provide estimates for them. In particular, we prove the existence of non-diagonal critical sections,
and establish inequalities whose combinatorial counterparts were proved earlier by Lesieur and Nicolas. Our approach is purely combinatorial, and it is based on recursive properties of the Laplace-Pólya integral,
which stands in stark contrast with previous results which apply involved analytical estimates.
This is a joint work with Barnabás Gárgyán.
Imre Barany, Alfréd Rényi Mathematical Institute, Hungary, and University College London, Great Britain
A matrix version of the Steinitz lemma
Abstract: The Steinitz lemma, a classic from 1913, states that a_1,...,a_n, a sequence of at most unit vectors in R^d with sum_1^n a_i=0, can be rearranged so that every partial sum of the rearranged sequence has norm at most 2d. It is an important result with several applications. In the matrix version of the Steinitz lemma A is a k by n matrix with entries a_i^j (at most unit vectors again) in R^d with sum a_i^j=0. Oertel, Paat, Weismantel have proved recently that there is a rearrangement of row j of A (for every j) such that the sum of the entries in the first m columns of the rearranged matrix has norm at most 40d^5 (for every m).
We improve this bound to 4d-2.
Sergey Bobkov, University of Minnesota, United States
Quantified Cramer-Wold continuity theorem for the Kantorovich transport distance
Abstract: An upper bound for the Kantorovich transport distance between probability
measures on multidimensional Euclidean spaces is given in terms of transport distances
between one dimensional projections. This quantifies the Cramer-Wold continuity theorem
for the weak convergence of probability measures. Joint work with F. Goetze.
René Brandenberg, Technical University of Munich, Germany
Recent results on p-means of convex bodies
Abstract: Firey was the first to define p-means of convex bodies and demonstrate their ordering by set containment. The p-means gained interest especially due to their connection to the log-Brunn-Minkowski conjecture and more recently by the extended effort in defining a proper geometric mean. In this talk, we summarize recent (re-)definitions and findings on the properties of p-means and their interrelations. We point out a somewhat surprising appearance of the golden ratio, applications to geometric inequalities, and the study of the Banach-Mazur distance.
Finally, we introduce a new geometric mean that balances simplicity with the natural properties expected in recent literature.
Leo Brauner, TU Wien, Austria
Mixed volumes and mixed area measures of bodies of revolution
Abstract: Sometimes, two mixed volume functionals with different reference bodies are in fact equal. For reference bodies that are symmetric around an axis, we determine when this is the case, using mixed spherical projections and tools from valuation theory. As an application, for convex bodies of revolution, we provide a complete solution to Christoffel-Minkowski type problems of intermediate degrees.
Christian Buchta, Salzburg University, Austria
Numbers of Vertices and Volumes of Random Polytopes
Abstract: Consider $n$ random points distributed independently. The convex hull of these points is a random polytope. Since the sixties of the last century relations between the numbers of vertices and the volumes of random polytopes have been studied. These relations turn out to be particular cases of a duality between the numbers of vertices and the volumes. The proof of the duality is based on a transformation formula for elementary symmetric polynomials.
Almut Burchard, University of Toronto, Canada
Symmetry-breaking in isodiametric capacitor problems
Abstract: It well-known that balls minimize the Newton capacity among bodies of given volume. On the other hand, Szegö proved that balls maximize the Newton capacity among bodies of given diameter.
While the lower bound on capacity in terms of volume is a cornerstone of potential theory, Szegö's upper bound in terms of diameter was largely relegated to a footnote.
Currently there is renewed interest in isodiametric shape optimization for Riesz capacities, in the context of aggregation problems with pair interactions of attractive-repulsive type.
In this talk I will discuss situations where the capacity-maximizer is not a ball. When does symmetry-breaking occur, and what are good candidates for non-radial maximizers?
Alexandros Eskenazis, CNRS, Sorbonne Université, France
Concavity principles for weighted marginals
Abstract: We shall present a general framework to study concavity properties of weighted marginals of measures via local methods. Connections to the B-inequality and the dimensional Brunn-Minkowski inequality for measures
shall also be discussed. The talk is based on joint work with D. Cordero-Erausquin.
Dmitry Faifman, Université de Montréal, Canada
Stability in the Banach isometric conjecture for planar sections
Abstract: The Banach isometric conjecture asks whether a convex body, such that all its k-dimensional sections through a fixed interior point are linearly equivalent to one another, is in fact an ellipsoid. Despite much progress, which has seen many special cases positively resolved (k=2 by Auerbach-Mazur-Ulam, k even by Gromov), and particularly in recent years (k=1 mod 4 by Bor, Hernandez-Lamoneda, Jimenez-Desantiago and Montejano-Peimbert, k=3 by Ivanov-Mamaev-Nordskova), the problem remains open. We will discuss the corresponding stability question, particularly for k=2. Based on a joint work with Gautam Aishwarya.
Daniel Galicer, Universidad de Buenos Aires, Argentina
Projection constants for function spaces
Abstract: In the theory of normed spaces, it is well known that every finite-dimensional subspace is complemented in any superspace containing it.
Building on this, a natural question arises: how can the norm of a projection onto a given subspace be quantified?
A key concept to address this problem is the so-called projection constant of the space X, defined as the smallest constant c > 0 such that, for any superspace Y containing X, there exists a projection from
Y onto X with norm not exceeding c.
In this talk, we provide explicit formulas for the projection constants of several important function spaces and discuss methods for computing them or estimating their asymptotic growth as the dimension of the ambient space
tends to infinity.
This is based on work done in collaboration with A. Defant, M. Mansilla, M. Masty lo, and S. Muro.
Andrii Ilienko, University of Bern, Switzerland, and Igor Sikorsky Kyiv Polytechnic Institute, Ukraine
Integer-valued convex valuations
Abstract: Valuations are a classical topic in convex geometry. Most studies have focused on continuous valuations invariant under specific transformation groups.
For instance, according to Hadwiger's classical theorem, any continuous valuation invariant under rigid motions can be expressed as a linear combination of intrinsic volumes.
A broader class of translation-invariant valuations has also been extensively studied, but their complete classification remains an open and possibly unsolvable problem.
In this talk, we take a different approach to the study of valuations. Instead of assuming invariance under a transformation group, we focus on valuations that take nonnegative integer values.
Such valuations are of particular interest in stochastic geometry and related fields.
We provide a full characterization of monotone sigma-continuous integer-valued valuations on the plane without imposing any kind of invariance with respect to motions. Additionally, we explore the consequences
of relaxing either the continuity or monotonicity conditions and discuss the challenges encountered when extending these results to higher-dimensional spaces.
This talk is based on an ongoing joint project with Ilya Molchanov and Tommaso Visonà.
Grigory Ivanov, PUC-Rio, Brazil
John's inclusion for log-concave functions
Abstract: John's inclusion states that a convex body in R^d can be covered by the d-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions.
As a byproduct of our approach, we establish the following asymptotically tight inequality: For any log-concave function f with finite, positive integral, there exist a positive-definite matrix A, a point a in R^d, and a positive constant alpha such that
X_B(x) <= alpha f(A(x-a)) <= \sqrt{d+1} e^{-|x|/(d+2) + (d+1)},
where X_B is the indicator function of the unit ball B in R^d.
Benjamin Jaye, Georgia Institute of Technology, United States
The mobile sampling problem
Abstract: Given a convex set K,
we will discuss the problem of how dense a surface needs to be to uniquely reconstruct a function whose Fourier transform is supported in K by its values on the surface.
Joint work with Mishko Mitkovski and Manasa Vempati.
Gil Kur, ETH Zürich, Switzerland
On the Role of Gaussian Covariates in Minimum Norm Interpolation
Abstract: In the literature on benign overfitting in linear models, also referred to as minimum norm interpolation, it is typically assumed that the covariates follow a Gaussian distribution. Existing proofs heavily rely on the Gaussian Minimax Theorem (GMT), making them inapplicable to other distributions in the linear setting. In our work, we are the first to establish matching rates for sub-Gaussian covariates in $\ell_p$-linear regression through a novel approach inspired by modern functional analysis. In this talk, we provide an overview of this proof and explore the role of Gaussian covariates in benign overfitting from a purely geometric perspective.
Zsolt Lángi, Budapest University of Technology and Economics and Alfréd Rényi Institute of Mathematics, Hungary
Spherical Steiner Symmetrization
Abstract: The aim of this talk is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbolic space introduced by Peyerimhoff in 2002. We show that this symmetrization preserves volume in every dimension, and investigate when it preserves convexity.
In addition, we examine the monotonicity properties of the perimeter and diameter of a set under this process, and find conditions under which the image of a spherically convex disk under a suitable sequence of Steiner symmetrizations converges to a spherical cap.
We talk about applications of our method to prove a spherical analogue of a theorem of Sas, and to confirm a conjecture of Besau and Werner about spherical floating bodies for centrally symmetric spherically convex disks. We also describe a spherical variant of a theorem of Winternitz. Joint work with Bushra Basit, Steven Hoehner and Jeff Ledford.
Alexander Litvak, University of Alberta, Canada
Rademacher projection and MM^*-estimate in the non-symmetric case.
Abstract: Let K be a centrally symmetric n-dimensional convex body and d_K denote the Banach-Mazur distance from K to the Euclidean ball.
The Pisier bound on the norm of Rademacher projection R from L_2(K) to L_2(K) of order log(d_K) was a major tool in obtaining a logarithmic bound on MM^*(K).
Later it was shown that a properly adjusted notion of the norm of Rademacher projection can be still used to bound MM^*(K) in the absence of symmetry and that this norm is bounded above
by \sqrt(d_K). We prove the sharpness of this bound by constructing a high-dimensional convex body with a large norm of Rademacher projection but logarithmically small MM^*.
This is a joint work with F. Nazarov.
Galyna Livshyts, Georgia Institute of Technology, United States
Weighted Blaschke-Santalo-type inequalities
Abstract: The classical Blaschke-Santalo inequality asserts that the volume product of a symmetric convex body is maximized when it is a ball.
All previously known functional analogues and extensions of this powerful inequality have been rotationally invariant.
In this talk, we discuss a new framework which allows for non-round maximizers in an analogue of this isoperimetric-type phenomenon.
Our study is linked also to strengthenings of the Brascamp-Lieb inequality for a class of measures, in the case of even functions.
Based on the joint work with Colesanti, Kolesnikov, Rotem.
Efrén Morales-Amaya, Autonomous University of Guerrero, Mexico
Characterizations of Special Convex Bodies in Terms of Support Cones
Abstract: A classical problem in convexity is to determine properties of a convex body K, i.e. a compact and convex set with non-empty interior, in the Euclidean space R^n, n>=3, from the information of its orthogonal projections.
For instance, in dimension 3 one can prove that if all the orthogonal projections of a body K are circles, then K is a Euclidean ball.
One can see this problem from the following perspective: consider the family of cylinders circumscribed to K and impose for each of them a condition in the section which is obtained with a hyperplane perpendicular to the lines generating the cylinder.
In our example, this means that we have a convex body K, subset R^3, such that for every cylinder Omega circumscribed to K, the section of H with Omega is a circle, where H is a plane orthogonal to the lines generating Omega.
Since cylinders are cones with apexes at the infinity, the original problem mentioned above can be formulated in the following manner: Determine properties of convex bodies imposing conditions on the sections of cones circumscribed to K whose apexes are in a fixed hyperplane.
Naturally, we can replace in this problem the condition that the set of apexes is a subset of a hyperplane by the condition that they are in a hypersurface S.
In particular, we can assume that S is the boundary of a convex body M subset of R^n such that K is a subset of the interior of M. An interesting example of this type is the well known Matsuura's Theorem.
Let K be a subset of R^3 be a convex body and let S be a closed convex surface which contains K in its interior. If the support cone of K from every point in S is a right circular cone, then K is a Euclidean ball.
In this talk we are going to present some results in terms of geometrical properties of the support cones of a convex body which characterizes balls, ellipsoid, centrally symmetric bodies, etc.
Francisco Marín Sola, CUD San Javier, Spain
On mth-order inequalities for log-concave functions
Abstract: The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric inequalities that relate the volumes of a convex body to those of its difference body and polar projection body, respectively.
Following a classical work by Schneider, both inequalities have recently been extended to the so-called mth-order setting. In this talk, on the one hand, we will present the mth-order analogue of Zhang's inequality
for log-concave functions. On the other hand, we will discuss some possible generalizations of the functional Rogers-Shephard inequality to the mth-order case.
This is part of a joint work with Dylan Langharst and Jacopo Ulivelli.
Fernanda Helen Moreira Baêta, TU Wien, Austria
A Characterization of Functional Affine Surface Areas
Abstract: In this work, we introduce functional affine surface areas by classifying certain valuations on convex functions with compact domains in R^n. We also investigate the corresponding dual problem via the Legendre transform.
As an application, we present the asymptotic approximation of C^2 convex functions with compact domains by piecewise affine functions. This is a joint work with Monika Ludwig.
Mohamed Abdeldjalil Mouamine, TU Wien, Austria
Vector-valued valuations on convex functions
Abstract: Following the work of A.Colesanti, M.Ludwig, and F.Mussnig, who established a functional version of Hadwiger's theorem which characterizes functional intrinsic volumes on the space
of convex super-coercive functions, we explore a novel family of valuations. Specifically, we characterize continuous, R^n-valued, epi-translation invariant and rotation equivariant valuations
and show that they can be constructed using Hessian measures. These valuations are derived from a Steiner formula applied to a functional version of the moment vector. We also establish
integral geometric formulas for these valuations. This is a joint work with Fabian Mussnig.
Márton Naszódi, Alfréd Rényi Institute of Mathematics and Loránt Eötvös University, Hungary
Higher rank antipodality
Abstract: Motivated by general probability theory, we say that the set X in R^d is antipodal of rank k, if for any k+1 elements q_1, ..., q_{k+1} in X$,
there is an affine map from the convex hull of conv X to the k-dimensional simplex Delta_k that maps q_1, ..., q_{k+1} onto the k+1 vertices of Delta_k.
For k=1, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on
antipodal sets: What is the maximum size of an antipodal set of rank k in R^d? We present a geometric characterization of antipodal sets of rank k and adapting the argument of Danzer and Grünbaum
originally developed for the k=1 case, we prove an upper bound which is exponential in the dimension. We point out that this problem can be connected to a classical question in computer science on finding perfect
hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension.
Joint work with Zsombor Szilágyi and Mihály Weiner.
Oscar Ortega Moreno, TU Wien, Austria
The Klain Approach to Zonal Valuations
Abstract: In this talk, I will present an analogue of the Klain - Schneider theorem for valuations that are invariant under rotations around a fixed axis.
Using this result, we derive a new integral representation for these valuations involving mixed area measures with a disk.
Additionally, I will introduce a simple method to translate between this representation and the standard one involving area measures.
Finally, I will discuss further applications of these results in integral geometry. Joint work with L. Brauner and G. C. Hofstätter.
Luis Rademacher, University of California Davis, United States
Expected extremal area of facets of random polytopes
Abstract: We study extremal properties of spherical random polytopes, the convex hull of random points chosen from the unit Euclidean sphere in R^n.
The extremal properties of interest are the expected values of the maximum and minimum surface area among facets.
We determine the asymptotic growth in every fixed dimension, up to absolute constants.
Martin Rapaport, Carnegie Mellon University, United States
Some Analogies Between Information Theory and Convex Geometry
Abstract: This talk is in the line of research that explores the connections between information theory and convex geometry. We will begin by introducing the definitions of discrete entropy and differential entropy, and the motivations behind these concepts. We will then focus on the entropy power inequality (EPI), a key result in information theory. Next, we will introduce the notion of log-concavity in our framework.
Our work involves proving an EPI-type inequality in the discrete setting of Z^d, via the monotonicity of discrete entropy for log-concave random vectors. Finally, in another ongoing work, we will present some results around Bergström's inequality in the context of information theory. We will conclude with open questions. Based on joint works with Lampros Gavalakis and Matthieu Fradelizi.
Mia Runge, Technical University of Munich, Germany
A first complete Blaschke-Santaló diagram for three-dimensional convex bodies
Abstract: We present a complete (r,D,R)-Blaschke-Santaló diagram for the 1-norm in 3-space, which consists of exactly three boundary parts.
While two of these boundaries are based on well-known inequalities, we prove one new inequality and show that this one completes the description of the diagram.
Ádám Sagmeister, University of Szeged, Hungary
Width related problems in spaces of constant curvature
Abstract: We revisit two classical problems that can be both interpreted as reverse isodiametric inequalities: Pál's isominwidth inequality stating that the equilateral triangle has minimal area if the minimal width is fixed,
and the Blaschke-Lebesgue inequality stating that among bodies of constant width, the minimizers are Reuleaux triangles. We find a connection between these two problems through spindle convexity.
We also introduce the spherical and hyperbolic analogues of these problems, and obtain stability results. In most cases, the optimal convex bodies for these problems are reduced,
i.e. they cannot be truncated without decreasing their minimal width; so we discuss reducedness in these spaces. Finally, we will talk about some recent results on reduced polygons.
The talk is partially based on joint works with Károly Jr. Böröczky, András Csépai, Ferenc Fodor and Ansgar Freyer.
Carsten Schütt, Christian-Albrechts-Universität zu Kiel, Germany
Banach-Mazur distance between \ell_p^n \otimes_\epsilon \ell_q^n and the spaces \ell_1^{n^{2}}, \ell_2^{n^{2}}
and \ell_{\infty}^{n^{2}}
Abstract: We are computing the Banach-Mazur distance between the spaces \ell_p^n \otimes \ell_q^n with 1 <= p,q <= \infty
and the spaces \ell_{1}^{n^{2}} and \ell_{\infty}^{n^{2}}.
Paul Simanjuntak, Texas A&M, United States
Empirical Methods for Dual Mixed Volumes
Abstract: The dual quermassintegrals of a star body are defined as the average volume of sections of the body by hyperplanes of fixed dimension.
These quantities, along with general dual mixed volumes, see striking parallels with their counterparts in Brunn-Minkowksi theory, especially for their respective extremal inequalities.
In this talk, we show how empirical methods provide a way to establish dual inequalities which are based on new local stochastic features within dual Brunn-Minkowski theory.
In particular, we show how symmetrization techniques feature again in the dual theory to prove extremal inequalities. Joint work with P. Pivovarov and G. Paouris.
Alina Stancu, Concordia University, Canada
On two results for the homothety conjecture in the plane
Abstract: This is a short talk on two results regarding the homothety conjecture in the plane which states, roughly, that the surface of flotation of a convex body is a scaling of the surface of K iff K is an ellipsoid :
a positive result when the convex body is centrally symmetric and close to a Euclidean ball, and a negative result when K is not centrally symmetric. This is joint work with M. Alfonseca, F. Nazarov, D. Ryabogin and V. Yaskin.
Jacopo Ulivelli, TU Wien, Austria
Geometric inequalities for convex functions
Abstract: Geometric inequalities for convex bodies are often strongly related to functionals that can be classified as sufficiently regular valuations.
In the functional setting, the same intuition fails, as we will show in this talk.
Nonetheless, it is still possible to identify cones of convex functions where inequalities can be recovered, leading to a unified geometric perspective and reinterpretation of results in the previous literature.
Based on joint works with Mussnig and Knoerr.
Elisabeth Werner, Case Western Reserve University, United States
The L_p-Floating Area, Entropy, and Isoperimetric Inequalities on the Sphere
Abstract: The floating area was previously investigated as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant positive curvature.
We introduce the family of L_p-floating areas for spherical convex bodies, as an analog to L_p-affine surface area measures from Euclidean geometry.
We investigate a duality formula, monotonicity and isoperimetric inequalities for this new family of curvature measures on spherical convex bodies.
Furthermore, using the L_p-floating area, we introduce a new entropy functional for spherical convex bodies and a dual isoperimetric inequality is established.
Based on joint works with Florian Besau.
Bartlomiej Zawalski, Kent State University, United States
On convex bodies with symmetric sections and an aligned center of (affine) symmetry
Abstract: Among all convex bodies, sections of ellipsoids and bodies of revolution exhibit particular symmetry.
Namely, all hyperplanar sections of an ellipsoid are centrally symmetric and have an axis of symmetry,
whereas all hyperplanar sections of a body of revolution have an axis of revolution. H. Brunn proved in 1889 that
the central symmetry of all the sections characterizes ellipsoids. Much later, in 1965, C.A. Rogers observed that it is enough
to consider only sections passing through a fixed point. Regarding axial symmetry, in 1999, K. Bezdek posed
his celebrated conjecture that the axial symmetry of all the sections characterizes bodies of revolution in 3-dimensional space.
Now, it is natural to formulate higher-dimensional analogs of Bezdek's conjecture, and there are many ways to do it.
Our main result is a proof of a certain variant of Bezdek's conjecture in arbitrary dimension n >= 3, where we assume that
all the sections passing through a fixed point have an axis of symmetry, satisfying certain alignment condition.
Further, if we weaken the hypothesis and consider only a 1-codimensional family of hyperplanes,
we obtain a similar characterization of axially symmetric bodies. For each of these problems, we show both the orthogonal
and affine variants. Interestingly, in 3-dimensional space the proof is essentially different and touches on the theory of
floating bodies. The talk is based on a joint work in progress with M. Angeles Alfonseca.
