April 17, 2014

September 13-17, 2010
Workshop on Asymptotic Geometric Analysis and Convexity

Talk Titles and Abstracts

Random matrices with independent log-concave rows
Radoslaw Adamczak
University of Warsaw / Fields Institute
Coauthors: O. Guedon, A. Litvak, A. Pajor, N. Tomczak-Jaegermann

I will discuss several properties of random matrices with independent log-concave rows, obtained during the last several years, including estimates on their norms, a solution to the Kannan-Lovasz-Simonovits problem and estimates on their smallest singular value. If time permits I will also briefly mention some limiting results for their spectral distributions.


Volume of Lp-zonotopes and best best constants in Brascamp-Lieb inequalities
David Alonso
Universidad de Zaragoza, Fields Institute

Given some unit vectors a1, ..., am ? Rn that span all Rn and some positive numbers q1, ..., qm, we consider for every p = 1 the convex body

Kp:={ x ? Rn : ?i=1m |<x, [(ai)/(qi)]>|p = 1}.

We will give some upper bounds for the volume of Kp and some lower bounds for the volume of its polar, depending on some parameters, which improve the ones obtained using the Brascamp-Lieb inequality. We will also see how the best choice of this parameters is related to the transformation which takes Kp to a special position which, for instance, when p=8, is John's position.


Gaussian and almost Gaussian formulas for volumes and the number of integer points in polytopes
Alexander Barvinok
University of Michigan
Coauthors: John A. Hartigan (Yale)

We present a family of computationally efficient formulas for volumes and the number of integer points in polytopes represented as the intersection of the non-negative orthant and an affine subspace. Although the formulas are not always applicable, they are asymptotically exact in a wide variety of situations. In particular, we obtain asymptotic formulas for the number of non-negative integer matrices with prescribed row and column sums and for the volumes of the respective transportation polytopes. The intuition for the formulas is provided by the maximum entropy principle, the Local Central Limit Theorem and its ramifications.


Invariant distributions in integral geometry
Gautier Berck

When the group, or the isotropy subgroup, is non-compact, classical Crofton type formulae may fail to exist because of the appearence of divergent integrals. The aim of the talk is to show that in some situations this problem may be circumvented replacing the invariant measure by an invariant distribution. The procedure will be illustrated by basic examples and applications in convex geometry.


On the location of roots of Steiner polynomials
Maria A. Hernandez Cifre
Departamento de Matematicas, Universidad de Murcia
Coauthors: Martin Henk

For two convex bodies K, E of the n-dimensional euclidean space and a non-negative real number x, the volume of K+xE is a polynomial of degree n in x, whose coefficients are, up to a constant, important measures associated to both sets, the relative quermassintegrals. This polynomial is called the (relative) Steiner polynomial of K (with respect to E). If we consider the Steiner polynomial as a formal polynomial in a complex variable, we are interested in studying geometric properties of its roots: their location in the complex plane, size, relation with other geometric magnitudes (in- and circumradius) and characterization of (families of) convex bodies by mean of properties of the roots. In this talk I will show the known results on this topic, which had its starting point in a problem posed by Teissier in 1982, in the context of Algebraic Geometry.


Integral functionals verifying a Brunn-Minkowski type inequality
Andrea Colesanti
Universita' di Firenze
Coauthors: Daniel Hug, Eugenia Saorin-Gomez

We consider a class of integral functionals defined in the family of convex bodies. The value of the functional on a convex body is given by the integral of a fixed continuous function defined on the unit sphere, with respect to the area measure of the convex body. We assume that a functional of this form verifies an inequality of Brunn-Minkowski type. We prove that if in addition the functional is symmetric, then it must be a mixed volume. The same result holds if the function defining the functional has some regularity property.


Small ball probability estimates, ?_2-behavior and the hyperplane conjecture

Nikos Dafnis
University of Athens

We introduce a method which leads to upper bounds for the isotropic constant. We prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small
probability estimates for the Euclidean norm on isotropic convex bodies. As a consequence of our method, we obtain an alternative proof of the result of J. Bourgain that every ?2-body has bounded isotropic constant, with a slightly better estimate.


The Poisson summation formula uniquely characterizes the Fourier Transform
Dmitry Faifman
Ph.D student, Tel-Aviv university

We show that under some regularity assumptions, The Poisson summation formula uniquely defines the Fourier Transform of a function. Then, we show how a family of unitary operators on L^2[0,infinity) can be constructed which exhibit Poisson-like summation formulas As a by-product of this construction, peculiar unitary operators given by series arise.


Concentration of measure phenomenon and eigenvalues of Laplacian
Kei Funano
Kumamoto university
Coauthors: Takashi Shioya (Tohoku university)

In this talk, we discuss the relation between the concentration of measure phenomenon and (behavior of) eigenvalues of Laplacian on a closed Riemannian manifold. M. Gromov and V. D. Milman was first studied for the case of the first non-trivial eigenvalue of Laplacian. Under non-negative Ricci curvature assumption we study the case of the k-th eigenvalues of Laplacian for any k.


Embedding from lpn into lrN for 0 < r < p < 2.
Olivier GUEDON
Université Paris-Est Marne-La-Vallée
Coauthors: Omer FRIEDLAND

We will present a Kashin type result for embedding lpn into l1N for 1 < p < 2 and N arbitrarily close to n. We will show that it is possible to define random embedings such that the conclusion holds with overwhelming probability. The result can also be extended to embedding from lpn to lrN with 0 < r = 1. One of the main tool that we developp is a new type of multi-dimensional Esseen inequality for studying small ball probabilities.


Minkowski valuations intertwining the special linear group
Christoph Haberl
Vienna University of Technology

A classification of SL(n) co- or contravariant Minkowski valuations will be presented. Here, a Minkowski valuation is a mapping from convex bodies to convex bodies which satisfies the inclusion-exclusion principle. Thereby, we obtain new characterizations of the projection and centroid body operator. Our result shows that the additional assumption of homogeneity in previous classifications is not necessary.


The average Frobenius number
Martin Henk
University of Magdeburg
Coauthors: Iskander Aliev, Aicke Hinrichs

Given a primitive positive integer vector a ? Zn > 0, the largest integer that cannot be represented as a non-negative integer combination of the coefficients of a is called the Frobenius number of a. In a series of papers V.I. Arnold initiated the research to study the average size of Frobenius numbers, and in a recent paper, Bourgain and Sinai showed that the probability of a "large" Frobenius number is "comparable small". Based on an approach using methods from Geometry of Numbers we can strengthen this result in such a way that we can estimate the average size of Frobenius numbers. Together with a discrete version of a reverse arithmetic-geometric-mean inequality by Gluskin and Milman, this allows us to show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bound, which, in particular, strengthens a recent result of Marklof on the asymptotic distribution of Frobenius numbers. Furthermore, we discuss generalizations to the case of more than one input vector a.


Volume and mixed volume inequalities in stochastic geometry
Daniel Hug
Karlsruhe Institute of Technology
Coauthors: Karoly Böröczky, Rolf Schneider, ...

Stochastic geometry deals with random structures such as random closed sets, random processes of flats or random tessellations. A useful method for analyzing such structures is to associate a deterministic convex set (sometimes a zonoid) with it. Thus strong results from convex geometric analysis become available. As appetizers, we give two examples:

Let Z0 denote the zero cell of a stationary Poisson hyperplane tessellation. We are interested in sharp bounds for the expected number of vertices of Z0. Such bounds are provided by the Blaschke-Santaló inequality and by the Mahler inequality for zonoids. Equality cases in these bounds characterize special direction distributions of the given hyperplane tessellation. Recently, these bounds have been improved by corresponding stability estimates, first in the geometric and then in the probabilistic framework. (joint work with Károly Böröczky)

As a second, new example, let X denote a stationary Poisson hyperplane process with fixed intensity g in \Rn. From X we pass to the intersection process X(k) of order k, which is a stationary process of (n-k)-flats in \Rn. It is well known that the intersection density g(k)(X), i.e. the intensity of X(k), is maximal if and only if X is isotropic. Here we introduce a measure for the strength of intersections from an affine-invariant point of view. The problem of determining its minimal value leads to a novel geometric inequality for mixed volumes of zonoids with isotropic generating measures. The solution of a related problem involving joint intersections from a process of lines and an independent process of hyperplanes is partly based on Keith Ball's reverse isoperimetric inequality together with the equality conditions due to Franck Barthe. (joint work with Rolf Schneider)


On the extremal distance between two convex bodies
C Hugo Jimenez
University of Seville
Coauthors: Márton Naszódi Eötvös University, Budapest

We consider d(K, L) a modified version of the Banach-Mazur distance of convex bodies in Rn proposed by Grünbaum. Gordon, Litvak, Meyer and Pajor in 2004 showed that for any two convex bodies d(K, L) = n, moreover, if K is a simplex and L=-L then d(K, L)=n. The following question arises naturally: Is equality only attained when one of the sets is a simplex? Leichtweiss, and later Palmon proved that if d(K, B2n)=n, where B2n is the Euclidean ball, then K is the simplex. We prove the affirmative answer to the question in the case when one of the bodies is strictly convex or smooth, thus obtaining a generalization of the result of Leichtweiss and Palmon.


If you can hide behind it, can you hide inside it?
Dan Klain
University of Massachusetts Lowell

Suppose that K and L are compact convex subsets of Euclidean space, and suppose that, for every direction u, the orthogonal projection (that is, the shadow) of L onto the subspace u? normal to u contains a translate of the corresponding projection of the body K. Does this imply that the original body L contains a translate of K? Can we even conclude that K has smaller volume than L?

In other words, suppose K can "hide behind" L from any point of view (and without rotating). Does this imply that K can "hide inside" the body L? Or, if not, do we know, at least, that K has smaller volume?

Although these questions have easily demonstrated negative answers in dimension 2 (since the projections are 1-dimensional, and convex 1-dimensional sets have very little structure), the (possibly surprising) answer to these questions continues to be No in Euclidean space of any finite dimension.

In this talk I will give concrete constructions for convex sets K and L in n-dimensional Euclidean space such that each (n-1)-dimensional shadow of L contains a translate of the corresponding shadow of K, while at the same time K has strictly greater volume than L. This construction turns out to be sufficiently straightforward that a talented person could conceivably mold 3-dimensional examples out of modeling clay.

The talk will then address a number of related questions, such as: under what additional conditions on K or L does shadow covering imply actual covering? What bounds can be placed on the volume ratio of K and L if the shadows of L cover those K?


The chain rule as a functional equation
Hermann Koenig
University of Kiel, Germany
Coauthors: S. Artstein-Avidan, V. Milman

Let T be an operator from C^1(R) into C(R) satisfying the chain rule functional equation T ( f o g) = (T f) o g * (T g) . We show that in a non-degenerate case any solution of this equation has the form (T f)(x) = H (f(x)) / H(x)* |f'(x)|^p * {sgn(f'(x)) , where H is continuous and p > 0 and where the last term sgn(f'(x)) may be missing; then also p = 0 is possible. An "initial condition" like T(2*Id) = 2 will imply that T f = f' holds. We also consider T operating on smoother functions C^k(R) or C^inf(R) and n-dimensional generalizations of the chain rule equation.


Moments of unconditional logarithmically concave vectors.
Rafal Latala
University of Warsaw

Let X=(X_1, X_2, .., X_n) be a random vector with unconditional logaritmically concave distribution. We will discuss several results and open problems related to moments of linear combinations of X_i's.


Rational Ehrhart quasi-polynomials
Eva Linke
Otto-von-Guericke-Universität Magdeburg

Ehrhart's famous theorem states that the number of integral points in a rationalpolytope is a quasi-polynomial in the integral dilation factor. We present the generalization to rational dilation factors. The number of integral points can still be written as a rational quasi-polynomial. Furthermore, the coefficients of this rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation.


On the Euclidean metric entropy.
A. Litvak
University of Alberta
Coauthors: V. Milman, A. Pajor, N. Tomczak-Jaegermann

We discuss some properties of entropy and covering numbers. In particular we show extension and lifting properties. We provide applications as well.


Another observation about operator compressions
Elizabeth Meckes
Case Western Reserve University
Coauthors: Mark Meckes

Let T be a self-adjoint operator on a finite-dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of T to a subspace of a given dimension is almost the same for almost all subspaces. This is a coordinate-free analogue of a recent result of Chatterjee and Ledoux on principal submatrices. The proof is based on measure concentration and entropy techniques, and the result improves on the result of Chatterjee and Ledoux in various ways. This is joint work with Mark Meckes.


Functional inequalities related to Mahler's conjecture
Mathieu Meyer
Université de Paris Est Marne-la-Vallée

We develop topics related to the title.


Properties of isoperimetric, spectral-gap and log-Sobolev inequalities via concentration
Emanuel Milman
University of Toronto

Various properties of isoperimetric, Sobolev-type and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is (possibly negatively) bounded from below.

First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L8 bound on the ratio between their densities, Total-Variation, Wasserstein distances, and relative entropy (Kullback-Leibler divergence). In particular, an extension of the Holley-Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic.

Next, in the compact setting, an optimal (up to numeric constants) isoperimetric inequality is obtained as a function of the curvature lower bound and diameter upper bound. In particular, the best known log-Sobolev inequality is obtained in this setting.

Time permitting, we will also mention the equivalence of Transport-Entropy inequalities with different cost functions and some of their applications.

The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting.


Hormander's proof of the Bourgain-Milman theorem
Fedor Nazarov
University of Wisconsin at Madison

A long standing Mahler's conjecture asserts that the product of the volumes of a symmetric convex body in Rn and its polar body is never less than Pn=4n/n!. Bourgain and Milman proved the lower bound cn Vn with some small positive constant c. Later, Kuperberg showed that one can take c=p/4. We shall use Hormander's ideas to give a fairly simple complex-analytic proof of the Bourgain-Milman theorem.


Feige's inequality
Krzysztof Oleszkiewicz
Institute of Mathematics: University of Warsaw & Polish Academy of Sciences

Let S denote a sum of (finitely many) independent non-negative random variables with means not exceeding 1. A remarkable result of Uriel Feige (SIAM Journal on Computing, 2006) states that for every 0<t<1 the inequality P(S<ES+t) > Ct holds true, where C is some universal positive constant (i.e. it does not depend on t, distributions and number of the random variables).

A short and simple proof will be presented, to a large extent along the lines of He, Zhang and Zhang (Mathematics of Operation Research, 2010), as well as some generalizations of the result.


Properties of metric spaces which are not coarsely embeddable into a Hilbert space
Mikhail Ostrovskii
St. Johns University, Queens, NY

The talk is devoted to expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained characterization can be used, for example, to derive the fact that infinite locally finite graphs excluding a minor embed coarsely into a Hilbert space.


On the existence of subgaussian directions for log-concave measures
Grigoris Paouris
Texas A&M University
Coauthors: A. Giannopoulos and P. Valettas

We show that if m is a centered log-concave probability measure on Rn then, [(c1)/(vn)] = |Y2(m)|1/n = c2[(v{logn})/(vn)], where Y2(m) is the y2-body of m, and c1, c2 > 0 are absolute constants. It follows that m has "almost subgaussian" directions: there exists q ? Sn-1 such that m({ x ? Rn : |<x, q>| = c t E |<·, q>| } ) = e- [( t2)/(log(t+1))] for all 1 = t = v{nlogn}, where c > 0 is an absolute constant.


On the reconstruction of inscribable sets in discrete tomography
Carla Peri
Università Cattolica - Piacenza
Coauthors: Paolo Dulio

In the usual continuous model for tomography one attempts to reconstruct a function from a knowledge of its line integrals. All the reconstruction methods used in computerized tomography require a large number of projection images to obtain results of acceptable quality.

The field of discrete tomography focuses on the reconstruction of samples, that consist of only a few different materials, from a "small" number of projections. For instance, it can be applied to the reconstruction of nanocrystals at atomic resolution, where it is assumed that the crystal contains only a few types of atoms, and that the atoms lie on a regular grid, modeled by the integer lattice. The high energies required to produce the discrete X-rays of a crystal mean that just a few number of X-rays can be taken before the crystal is damaged, so that the conventional techniques of computerized tomography fail.

In general, this reconstruction task is a ill-posed inverse problem. In fact, for general data there need not exist a solution, if the data is consistent, the solution need not be uniquely determined and "small" changes in the data can lead to unique but disjoint solutions. Thus, one has to use a priori information, such as convexity or connectedness, about the sets that have to be reconstructed to satisfy existence, uniqueness and stability requirements.

By now there are many uniqueness results available for different classes of finite lattice sets, but just few stability results. In the present paper we introduce some new classes of lattice sets, and investigate the problem of their reconstruction by means of their X-rays taken in the directions belonging to given finite set D. The geometric structure of such sets enable us to prove results concerning additivity and uniqueness. When D is the set of coordinate directions, we give a sharp stability estimate which depends only on the data error, differently from all the known results, which also involve the sizes of the sets. Some of these results hold true in any dimension.


On the volume of random convex sets
Peter Pivovarov
Fields Institute
Coauthors: Grigoris Paouris

Let K ? Rn be a convex body of volume one. Let X1, ..., XN be independent random vectors distributed uniformly in K and let KN be their (symmetric) convex hull. A result of Groemer's states that the expected volume of KN is smallest when K is the Euclidean ball of volume one. A similar result, due to Bourgain, Meyer, Milman and Pajor, holds for the volume of random zonotopes ZN=?i=1N Xi. If T:RN?Rn is the (random) linear operator defined by Tei=Xi, for i=1, ..., N, then KN is the image of the unit ball in l1N, while ZN is the image of the unit ball in l8N. What happens when T is applied to other sets? I will discuss a unified approach to various inequalities involving the volume of random convex sets for which the Euclidean ball is the minimizer.


Poisson-Voronoi approximation
Matthias Reitzner
Univ. Osnabrueck
Coauthors: Matthias Schulte

Let X be a Poisson point process and K a convex set. For a point x in X denote by v(x) the Voronoi cell with respect to X, and by vX (K) the union of all Voronoi cells with center in K. We call vX(K) the Poisson-Voronoi approximation of K.

For K a compact convex set the volume difference Vd(vX(K))-Vd(K) and the symmetric difference Vd(vX(K) \triangle K) are investigated. Estimates for the variance and central limit theorems are obtained using the chaotic decomposition of these functions in multiple Wiener-Ito integrals. (Work in progress jointly with Matthias Schulte)


Spectral properties of random conjunction matrices
by Mark Rudelson
University of Missouri
Coauthors: Shiva Kasiviswanathan, Adam Smith, and Jon Ullman

We consider a problem in random matrix theory, which arises from computer science. The standard way to release the statistical summary of the information contained in a large data base is to publish its contingency table, which contains percentages of records having several given common entries. However, if the contingency table is released exactly, one can reconstruct the individual entries by solving a system of equations. The standard way to protect the privacy of individual records is to add a random noise to the contingency table. Determining the minimal necessary amount of such noise leads to the problem of estimating the smallest singular value of a special random matrix with dependent entries, which is generated from a random matrix with i.i.d. entries taking values 0 and 1.


Translation Invariant Valuations
Franz Schuster
Vienna University of Technology
Coauthors: Semyon Alesker and Andreas Bernig

As a generalization of the notion of measure, valuations on convex bodies have long played a central role in geometry. The starting point for many important new results in valuation theory is Hadwiger's remarkable characterization of the continuous rigid motion invariant real valued valuations as linear combinations of the intrinsic volumes. Among many applications, this result allows an effortless proof of the famous Principal Kinematic Formula from integral geometry.

In the first part of this talk, the decomposition of the space of continuous translation invariant valuations into a sum of SO(n) irreducible subspaces is presented. It will be explained how this result can be reformulated in terms of a Hadwiger type theorem for translation invariant and SO(n) equivariant valuations with values in an arbitrary (finite dimensional) SO(n) module. From this perspective the classical theorem of Hadwiger becomes the special case when the SO(n) module is the trivial 1-dimensional one.

A striking recent development in valuation theory explores the connections between isoperimetric inequalities and convex body valued valuations. To be more specific, many powerful geometric inequalities involve fundamental operators on convex bodies which are valuations, e.g. projection and intersection body maps. In many instances the proofs of these inequalities are based on the symmetry of certain bivaluations associated with convex body valued valuations.

In the second part, the decomposition of the space of translation invariant valuations into irreducible SO(n) modules is used to study the symmetry of O(n) invariant bivaluations and to establish new Brunn-Minkowski type inequalities for convex body valued valuations.


Affine Differential Invariants for Convex Bodies
Alina Stancu
Concordia University, Montreal
Coauthors: n/a

In what concerns the affine component of Felix Klein’s Erlangen program, the first results were due to Blaschke’s school. They were of differential geometric nature and required at least C^4 regularity of closed convex hypersurfaces with, often, positive Gauss curvature everywhere. While this is unsatisfactory for the general study of convex bodies, certain objects – the most famous one being the affine surface area – have appeared in affine differential geometry but they were later extended to arbitrary convex bodies with suprising applications. We want to motivate a certain direction of research which seeks new affine differential invariants and their applications in view of possible extensions to arbitrary convex bodies. The talk will not require any prerequisite.


Estimation of covariance matrices
Roman Vershynin
University of Michigan

Estimation of the covariance matrix of a p-dimensional probability distribution is a basic problem in statistics. A classical estimator is the sample covariance matrix, obtained from a sample of n independent points. The more classical regime and well studied regime is where n > p. We conjecture that n = O(p) suffices to accurately estimate the covariance matrix of arbitrary distribution with finite 4-th moments. We discuss some recent progress on this problem, which has a connection to the "Levy flight", a heavy-tailed Brownian motion that exhibits sporadic huge jumps (similar to a predator's path looking for prey). The other regime, n < p, has recently become quite popular in statistics and its various applications (e.g. genomics) because of limiting sampling capacities compared with huge dimensions. We will discuss the problem and recent progress in this regime as well.


GL(n) intertwining Minkowski valuations
Thomas Wannerer
Vienna University of Technology
Coauthors: Franz E. Schuster

Recently, M. Ludwig characterized continuous Minkowski valuations which intertwine the general linear group under the assumption that the valuations either are defined on the set of convex bodies containing the origin or are invariant under translations. We extend these results without any restrictions on the domain or invariance under translations. Part of this work is joint with Franz Schuster.


Valuations and local functionals
Wolfgang Weil
Karlsruhe Institute of Technology

For the classical motion invariant valuations on convex bodies, the intrinsic volumes, local variants exist, namely measure-valued additive and motion covariant functionals. These are the curvature measures. In view of applications in integral geometry, we study translation invariant functionals on convex bodies or convex polytopes which admit such a local version. We clarify the connection between these local functionals and valuations and also discuss whether valuations on convex polytopes which have certain continuouity properties allow an extension to all convex bodies.


Volume Integral Means of Holomorphic Mappings
Jie Xiao
Memorial University
Coauthors: Kehe Zhu

In this talk we discuss some geometric aspects of several complex variables via considering the integral means of holomorphic functions in the unit complex ball with respect to weighted volume measures, including Sobolev-type embedding and isoperimetric inequalities associated to holomorphic maps, log-convexity of the integral means, and weighted Ricci curvatures.


Towards an Orlicz Brunn-Minkowski theory
Deane Yang
Polytechnic Institute of NYU
Coauthors: Erwin Lutwak and Gaoyong Zhang

Recent extensions of the classical and L_p Brunn-Minkowski theory to an Orlicz Brunn-Minkowski theory are discussed.


The geometry of p-convex intersection bodies.
Vladyslav Yaskin
University of Alberta
Coauthors: J.Kim and A.Zvavitch

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. We provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.


Shadow boundaries and the Fourier transform

Maryna Yaskina
University of Alberta

We investigate the Fourier transform of homogeneous functions on $\mathbb R^n$ which are not necessarily even. These techniques are applied to the study of nonsymmetric convex bodies, in particular to the question of reconstructing convex bodies from the information about their shadow boundaries.


On the homothety conjecture
Deping Ye
The Fields Institute
Coauthors: E. Werner

Let K be a convex body in Rn and Kd its floating body. The homothety conjecture asks: "Does Kd=c K imply that K is an ellipsoid?" Here c is a constant depending on d only. We prove that the homothety conjecture holds true in the class of the convex bodies Bnp, 1 = p = 8, the unit balls of lpn; namely, we show that (Bnp)d = c Bnp if and only if p=2. We also show that the homothety conjecture is true for a general convex body K if d is small enough.


Some geometric properties of Intersection Body Operator.
Artem Zvavitch
Kent State University
Coauthors: Fedor Nazarov, Dmitry Ryabogin

The notion of an intersection body of a star body was introduced by E. Lutwak: K is called the intersection body of L if the radial function of K in every direction is equal to the (d-1)-dimensional volume of the central hyperplane section of L perpendicular to this direction.

The notion turned out to be quite interesting and useful in Convex Geometry and Geometric tomography. It is easy to see that the intersection body of a ball is again a ball. E. Lutwak asked if there is any other star-shaped body that satisfy this property. We will present a solution to a local version of this problem: if a convex body K is closed to a unit ball and intersection body of K is equal to K, then K is a unit ball.

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