Abstracts. Analytic and Probabilistic Techniques in Modern Convex Geometry November 7-9, 2015 University of Missouri-Columbia

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1 Abstracts Analytic and Probabilistic Techniques in Modern Convex Geometry November 7-9, 2015 University of Missouri-Columbia Rotations of shadows of convex bodies: Positive Results and Counterexamples Maria de los Angeles Alfonseca-Cubero North Dakota State University We construct examples of two convex bodies K, L in R n, such that every projection of K onto a (n 1)-dimensional subspace can be rotated to be contained in the corresponding projection of L, but K itself cannot be rotated to be contained in L. We also find necessary conditions for K and L in R 3 to ensure that K can be rotated to be contained in L if all the 2-dimensional projections have this property. Second order concentration on the sphere Sergey Bobkov University of Minnesota The concentration of measure phenomenon on the unit sphere asserts that mean zero Lipschitz functions are of order at most 1/ n with a subgaussian domination for tails of their distribution functions. I will be discussing conditions on the derivatives of the functions that ensure the order 1/n for deviations with an exponential domination for tails. Joint work with G.P. Chistyakov and F. Göetze. Results surrounding Dvoretzky s theorem Daniel Fresen Weizmann Institute We discuss topics such as explicit subspaces, dependence on epsilon, and the way in which different Euclidean subspaces fit together. For spaces of dimension n with nontrivial cotype (i.e. bounded constant), it is easy to see that there exists a basis so that with respect to this basis, all sparse vectors behave as if they were in a Hilbert space i.e. satisfy Euclidean estimates with (1 + ɛ) distortion. What is more interesting is that this fails in l n (the phenomenon, not just the technique), even for sparsity 2. It may be possible that it holds in the isomorphic setting, but we do not know. We present a lemma that gives a simplified proof of the randomised isomorphic Dvoretzky theorem (going back to Milman/Schechtman, and also studied by Litvak/Mankiewicz/Tomczak-Jaegermann, and others). We also mention related (recent) work of Chasapis and Giannopoulos.

2 Program 2 Sums of sections, Fourier transforms and the Minkowski problem Paul Goodey University of Oklahoma In this talk I intend to survey some geometrical applications of the Fourier transform techniques pioneered by Alex Koldobsky. Over many years now, these techniques have shown themselves to be very successful in the resolution of long standing problems. Here I will describe some rather unexpected (at least to me) applications of the method which I hope demonstrate its effectiveness. My focus will be on problems involving bodies which are not necessarily centrally symmetric. In particular, I will try to show how two rather disparate topics, Berg s solution of the Christoffel problem and the study of Minkowski sums of sections of a given body, can be seen, via Fourier transform techniques, to be very closely related. The talk will be expository in nature and is based on joint works with Wolfgang Weil. Measure of sections of convex bodies Hermann Koenig Christian-Albrechts-Universitat We study various formulas for measures of sections of convex bodies, in particular for l n p-balls, which are related to work with A. Koldobsky. Let a S n 1 and A(a) := B n a n 1 be the Lebesgue measure of the hyperplane section. By a wellknown result of K. Ball, A(a) A( f 2 ) where f 2 := 1 (1, 1, 0,..., 0). We consider 2 the same problem for other types of product measures, absolutely continuous with respect to Lebesgue measure, in particular normalized Gaussian type measures on B, n exp( λ x 2 )dx. For small λ > 0, the result of K. Ball is true in this setting, too, but for larger λ it is false. We give a necessary and sufficient condition for certain types of product measures such that the analogue of Ball s result holds (A. Koldobsky, H. König). The proof uses Fourier analytic formulas for the measure and a tricky power type inequality for the Fourier transform of the density, which is shown using a technique of F. Nazarov and A. Podkorytov. We also consider generalizations of this result to lower dimensional sections of the cube. Another problem studied with A. Koldobsky is a question of Pelczynski on the maximal perimeter, i.e. (n 2)-dimensional measure of cubic sections. In the lecture, we also consider maximal or minimal sections of l n p-balls and the n-simplex. Entropy Numbers of Certain Integral and Summation Operators - The Critical Case Werner Linde University of Delaware We present some recent results about compactness properties of Volterra type operators V β defined by t f (x) (V β f )(t) = 0 (t x) 1/2 ln(t x) β dx. If β > 1/2, then V β is compact from L 2 [0, r] to L [0, r], r < 1. The entropy numbers e n (V β ) behave very differently in the regions 1/2 < β < 1 and 1 < β <. The critical case β = 1 remained open for some time and was finally solved by M. A. Lifshits in 2012.

3 Program 3 The operators V β are related to certain weighted summation operators S w on a tree T. If w : T [0, ) is a weight function, then (S w x)(t) = w(s)x(s), t T. s t If T is a binary tree, then the critical weight function is w(t) = t 1. And both problems are tightly connected with the following question: Let A be a compact subset of a Hilbert space. Knowing the behavior of e n (A) find suitable estimates of e n (aco(a)) where aco(a) denotes the absolutely convex hull of A. Here the critical behavior appears if e n (A) n 1/2. Invertibility of adjacency matrices of random digraphs Alexander Litvak We show that the adjacency matrix of a random d-regular directed graph on n vertices is invertible with probability at least 1 C ln 3 d/ d for C d cn/ ln 2 n (joint work with A. Lytova, K. Tikhomirov, N. Tomczak-Jaegermann and P. Youssef) On marginals of product densities Galyna Livshyts Georgia Tech It was shown by Ball that the maximal section of an n-dimensional cube is 2. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This talk is based on joint work with G. Paouris and P. Pivovarov. Expansion and anti-concentration properties for random d-regular digraphs Anna Lytova For random d-regular digraphs we establish certain expansion properties and also prove a Littlewood-Offord type anti-concentration property. Our analysis is essentially based on an operation called the simple switching. These properties play an important role in the proof of invertibility of adjacency matrices of random d-regular digraphs. A joint work with A. Litvak, K. Tikhomirov, N. Tomczak-Jaegermann, and P. Youssef. On a property of symmetric convex bodies and the low-dimensional Busemann-Petty Problem Emanuel Milman Technion - Israel Institute of Technology We investigate a certain new monotonicity property for symmetric convex bodies, and describe a surprising connection to the (yet unresolved) low-dimensional Busemann- Petty problem. Joint work with Alex Koldobsky.

4 Program 4 Algebraic Theory of Convex Bodies (after results of Liran Rotem) Vitali Milman University of Tel Aviv Approximating the covariance matrix Alain Pajor Universite de Paris Est Marne-la-Vallee We will discuss recent results on the rate of convergence of empirical covariance matrices. On convex bodies with congruent projections Dimitry Ryabogin Kent State University Let K and L be two convex bodies in R 4, such that their projections onto all 3- dimensional subspaces are directly congruent. Does it follow that K and L coincide up to translation and reflection in the origin? We prove that the answer is affirmative if the set of diameters of the bodies is not too large, and some of the projections do not have certain π-symmetries. Iterations of the projection body operator and a remark on Petty s conjectured projection inequality Christos Saroglou Kent State University We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence {Π m K} of convex bodies converges to the ball with respect to the Banach-Mazur distance, as m. Here, Π denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak. Polarity and Santaló point via cones Stanislaw Szarek Case Western Reserve University The purpose of this talk is to advertise an approach to polarity of not-necessarilysymmetric convex bodies via cone duality. In particular, this yields a simple proof of the well-known characterization of the Santaló point. All results are known, but not necessarily in all circles. The Floating Body in Real Space Forms Elisabeth Werner Case Western Reserve University The notion of convex floating body of a Euclidean convex body has recently been extended to the Euclidean unit sphere. We will present an extension to the hyperbolic

5 Program 5 setting and give a unifying approach. Thus a complete description of floating bodies in real space forms, that is, manifolds with constant curvature, is obtained. Differentiation of the volume of the floating body gives rise to the floating area. In the Euclidean setting the floating area is better known as affine surface area. It is a classical and powerful tool in the (equi-)affine geometry of convex bodies. The floating area can be seen as a real analytic extension of the affine surface area. This is joint work with Florian Besau. Uniqueness Questions in Geometric Tomography Vlad Yaskin We will discuss some results on the unique determination of convex bodies (and other objects) from various tomographic data. The dual Orlicz-Brunn-Minkowski theory Deping Ye Memorial University The discovery of Orlicz addition of convex bodies (by Gardner, Hug and Weil; and independently by Xi, Jin and Leng) makes it possible to better understand properties of the Orlicz-Brunn-Minkowski theory for convex bodies (initiated by Lutwak, Yang and Zhang). For instance, the Orlicz-Brunn-Minkowski inequality and the Orlicz-Minkowski inequality can be established with the help of Orlicz addition of convex bodies. In this talk, I will talk about the dual Orlicz-Brunn-Minkowski theory for star bodies. In particular, Orlicz addition for star bodies and related inequalities (such as the dual Orlicz-Brunn-Minkowski inequality and the dual Orlicz-Minkowski inequality) will be provided. With the appropriate choice of functions, we obtain the precise duals of the conjectured log-brunn-minkowski and log-minkowski inequalities of Boroczky, Lutwak, Yang and Zhang. Planar Sobolev extension domains a geometric description Nahum Zobin College of William and Mary A planar domain Ω is called a planar Sobolev (p, m)-extension domain, if any function from the homogeneous Sobolev space L m p (Ω) can be extended to a function from L m p (R 2 ). For p > 2, m 1 we obtain a complete geometric description of planar bounded simply connected Sobolev (p, m)-extension domains in terms of special subhyperbolic metrics on the domain. This resolves a long standing problem in the field. Joint work with P. Shvarstman.