Schedule for: 18w5081 - Emerging Trends in Geometric Functional Analysis

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The classical Rogers-Shephard inequality provides an upper bound of the volume of the difference body of an $n$-dimensional convex body $K$ and, more generally, it states that for any pair of convex bodies $K,L$ and any $x_0\in\R^n$ $$ |K\cap (x_0+L)||K-L|\leq{2n\choose n}|K||L|, $$ with equality if and only if $K=x_0+L$ is a simplex. A reverse inequality was given by Milman and Pajor, showing that for any pair of convex bodies with the same barycenter $$ |K||L|\leq|K-L||K\cap L|. $$ In this talk we will extend these inequalities to the more general setting of log-concave functions, showing that for any pair of integrable log-concave functions $f,g$, if $f\star g$ denotes their Asplund product and $f*g$ their convolution, we have that $$ \Vert f*g\Vert_\infty\Vert f\star g\Vert_1\leq {2n\choose n}\Vert f\Vert_\infty\Vert g\Vert_\infty\Vert f\Vert_1\Vert g\Vert_1, $$ with equality if and only if $\frac{f(x)}{\Vert f\Vert_\infty}=\frac{g(-x)}{\Vert g\Vert_\infty}$ is the characteristic function of a simplex and if $f$ and $g$ have opposite barycenters and attain their maximums at 0 $$ \Vert f\Vert_\infty\Vert g\Vert_\infty\Vert f\Vert_1\Vert g\Vert_1\leq e^{1+\textrm{Ent}\left(\frac{f}{\Vert f\Vert_\infty}\right)+\textrm{Ent}\left(\frac{g}{\Vert g\Vert_\infty}\right)} f*g(0)\Vert f\star g\Vert_1, $$ improving the value of the constant in some particular cases.

It is a classic result that the expected volume difference between a convex body and a random polytope, i.e., the convex hull of i.i.d. random points chosen uniformly from the convex body, converges to the affine surface area of the convex body as the number of points goes to infinity. Furthermore, if the convex body is actually a polytope, then the affine surface area vanishes and the first term in the asymptotic expansion of the volume difference depends on the number of (complete) flags of the polytope. Remarkably, a similar behavior is exhibited by the volume difference between a convex body and its floating body. In this talk I consider recent generalizations of the above notions obtained together with M.~Ludwig and E.~M.~Werner, where we consider the non-uniform case. This naturally gives rise to weighted floating bodies and a notion of weighted affine surface area for general convex bodies. In particular, from our results we can easily derive corresponding results for constant curvature spaces, i.e., the unit sphere or hyperbolic space. More recently, in joint work with C.~Sch{\"u}tt and E.~M.~Werner, we were able to give extensions also for the first term in the asymptotic expansion of the volume difference of a polytope and its weighted floating body, which now depends on weighted sum of the (complete) flags of the polytope. Our results raises the question, if again a similar behavior can be observed in the non-uniform random approximation of convex polytopes.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in $\mathbb{R}^n$ to the class of unit balls of $n$-dimensional subspaces of $L_p$-spaces. Finally, we prove a result of the Busemann-Petty type for these moments. This is a joint work with Sergey Bobkov and Bo'az Klartag.

A body $K$ is called polynomially integrable if its parallel section function $V_{n-1}(K\cap\{\xi^\perp+t\xi\})$ is a polynomial of $t$ (on its support) for every $\xi$. A complete characterization of such bodies was given recently. Here we obtain a generalization of these results in the setting of dual quermassintegrals. We also address the associated smoothness issues.

For a convex body $K \subset \R^n$, let $$K^z = \{y\in \R^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$$ be the polar body of $K$ with respect to the center of polarity $z \in \R^n$. In this talk we would like to discuss the maximum of the volume product $$\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K||K^z|,$$ among convex polytopes $K\subset {\mathbb R}^n$ with a number of vertices bounded by some fixed integer $m \ge n+1$. In particular, we will show that the supremum is reached at a simplicial polytope with exactly $m$ vertices and we provide a new proof of a result of Meyer and Reisner showing that, in the plane, the regular polygon has maximal volume product among all polygons with at most $m$ vertices.

Asymptotic results for weighted floating bodies of polytopes are established and applications to spherical and hyperbolic space are explored. Joint work with Florian Besau and Elisabeth M. Werner

In the recent years, a number of conjectures has appeared, concerning the improvement of the inequalities of Brunn-Minkowski type under the additional assumptions of symmetry; this includes the B-conjecture, the Gardner-Zvavitch conjecture of 2008, the Log-Brunn-Minkowski conjecture of 2012, and some variants. The conjecture of Gardner and Zvavitch, also known as dimensional Brunn-Minkowski conjecture, states that even log-concave measures in $\R^n$ are in fact $\frac{1}{n}$- concave with respect to the addition of symmetric convex sets. In this talk we shall establish the validity of the Gardner-Zvavitch conjecture asymptotically, and prove that the standard Gaussian measure enjoys $\frac{0.37}{n}$ concavity with respect to centered convex sets. Some improvements to the case of general log-concave measures shall be discussed as well. This is a joint work with A. Kolesnikov.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We show that a subspace of $\ell_\infty^N$ of dimension $n>(\log N\log\log N)^2$ contains $2$-isomorphic copies of $\ell_\infty^k$, where $k$ tends to infinity with $$n/(\log N\log \log N)^2.$$ More precisely, for every $\eta>0$, we show that any subspace of $\ell_\infty^N$ of dimension $n$ contains a subspace of dimension $m=c(\eta)\sqrt{n}/(\log N\log \log N)$ of distance at most $1+\eta$ from $\ell_\infty^m$. Some concrete estimates known for this dimension will be also discussed. Joint work with Gideon Schechtman.

In 2003 Cordero-Erausquin, Fradelizi, and Maurey proved that the standard Gaussian measure satisfies a certain concavity property known as the (B) property. This (B) property has two versions, a weak version and a strong one. It is conjectured that any even log-concave measure satisfies at least the weak (B)-property. We will show that strong (B) property can fail even for arbitrarily small Gaussian perturbations on the standard Gaussian measure. We will also discuss some results and equivalent formulations of the above conjecture regarding the weak (B) property. Based on joint work with D.~Cordero-Erausquin.

A simple way to generate a Boolean function in n variables is to take the sign of some polynomial. Such functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? This problem was solved for degree $d=1$ by Zuev in 1989 and has remained open for any higher degrees, including $d=2$, since then. In a joint work with Pierre Baldi (UCI), we settle the problem for all degrees $d>1.$ The solution explores connections of Boolean functions to additive combinatorics and high-dimensional probability. This leads to a program of extending random matrix theory to random tensors, which is mostly an uncharted territory at present.

Let $A = (a_{ij})$ be a square $n\times n$ matrix with i.i.d. zero mean and unit variance entries. In a paper by Rudelson and Vershynin it was shown that the upper bound for a smallest singular value $s_n(A)$ is of order $n^{-\frac12}$ with probability close to one under additional assumption on entries of $A$ that $\mathbb{E}a^4_{ij} < \infty$. We remove the assumption on the fourth moment and show the upper bound assuming only $\mathbb{E}a^2_{ij} = 1.$

The most classical problem in random matrix theory is to specify a natural joint distribution for the entries of a large random matrix, then study the asymptotic behavior of the distribution of the eigenvalues. I will describe joint work with Elizabeth Meckes on the opposite problem: For a natural model of random matrices with prescribed eigenvalues, we study the asymptotic behavior of the distribution of the matrix entries. Our results have applications to quantum mechanics, and shed new light on the universality phenomenon in classical random matrix theory.

A well known model in Stochastic Geometry is the Boolean model where balls are located at random in space and their union is investigated. The Vietoris-Rips and the Cech complex reflect the topological structure of the Boolean model. We are interested in concentration inequalities for the number of k-dimensional simplices of this simplicial complexes or more general subgraph counts. This gives rise to concentration inequalities for component counts. In the last part of the talk we investigate the Betti numbers of components of these simplicial complexes.

Let $P\subset\mathbb{R}^n$ $(n\geq 3)$ be a convex polytope containing the origin in its interior. Let $\mbox{vol}_{n-2} \big( \mbox{relbd} ( P\cap\lbrace t\xi + \xi^\perp \rbrace ) \big)$ denote the $(n-2)$-dimensional volume of the relative boundary of $P\cap\lbrace t\xi + \xi^\perp \rbrace$ for $t\in\mathbb{R}$, $\xi\in S^{n-1}$. We prove the following: if \begin{align*} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\cap\xi^\perp \big) \Big) = \max_{t\in\mathbb{R}} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\cap\lbrace t\xi + \xi^\perp \rbrace \big) \Big) \end{align*} for all $\xi\in S^{n-1}$, then $P = -P$. Our result gives a partial affirmative answer to a conjecture by Makai, Martini, and \'Odor.

n recent years, it was proven that there exist precisely four order isomorphisms acting in the class of geometric convex functions. These are the Legendre transform ${\cal L}$, the geometric duality transform ${\cal A}$, their composition ${\cal J}$, and the identity. It is known that ${\cal L}$ and ${\cal A}$ satisfy Santal\'{o}-type inequalities, e.g. the quantity $M(f) = {\rm Vol}(f)\cdot{\rm Vol}({\cal L}f)$ is bounded from above and below (here ${\rm Vol}(f)$ stands for the integral over ${\mathbb R}^n$ of $e^{-f}$). We prove similar (asymptotically sharp) bounds for the quantity $M^{\cal J} (f) = {\rm Vol}( {\cal J} f) / {\rm Vol}(f)$, and describe the extremal functions.

Analysis of condition number for random matrices originated in the works of von Neumann and Turing on complexity of linear system solving. For the case of non-linear algebraic equations (i.e. polynomials), there are two decoupling facts: feasibility of a system of equations over the complex numbers is NP-Hard, and a generic system of polynomials always has the same number of roots (i.e. Bezout number or BKK bound). These facts led to the search for algorithms that solves a generic polynomial system fast, and a notion of condition number arose naturally. The analysis of condition number for random polynomial systems thus became the central ingredient for understanding complexity of polynomial system solving. We present a rather quick survey on condition number analysis over complex numbers (i.e., a survey on the solution of Smale's 17th problem), then pass to the field of real numbers. We hope to finish by presenting our results on condition number analysis of random real polynomial systems. This is joint work with J. Maurice Rojas and Grigoris Paouris.

Let $C$ be a closed convex cone in $\mathbb{R}^n$, pointed and with interior points. A set $A = C \setminus K$, where $K \subset C$ is a closed convex set, is called a $C$-coconvex set if it has finite volume. The family of $C$-coconvex sets is closed under the addition $\oplus$ defined by $$C \setminus (A_1 \oplus A_2) = (C \setminus A_1) + (C \setminus A_2).$$ For compact $C$-coconvex sets, Khovanskii and Timorin (2014) have proved counterparts to the inequalities of the classical Brunn--Minkowski theory. For coconvex sets of finte volume, we prove a Brunn--Minkowski type inequality with equality discussion and then, as far as possible, counterparts to the uniqueness and existence theorems for sets with given surface area measures or cone-volume measures.

We discuss a new construction of bodies from a given convex body in $\mathbb{R}^n$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas.

In 2004 Klartag and Vershynin showed that Gaussian small-ball inequalities can be used to locate (one-sided) Euclidean structure in subspaces of a priori higher dimension than that suggested by V. Milman's formula. The purpose of this talk is to explain how we may combine the Gaussian convexity with superconcetration or hypercontractivity, to obtain strong small-ball estimates for norms. Based on a joint work with G. Paouris (Texas A\&M) and K. Tikhomirov (Princeton).

In recent work with Rafal Latala and Pierre Youssef, we were able to completely characterize the Schatten p-norms of random matrices with any independent centered Gaussian entries up to universal constants (independent of p and of the variance pattern of the entries). The result has a tantalizing probabilistic formulation: the distribution of the Schatten $p$-norm is comparable in distribution to the $l_p(l_2)$-norm of the rows and columns of the matrix. This simple probabilistic statement is however not reflected at all in the proof of the result, which is based on a combination of different kinds of analyses of the norms of random matrices by combinatorial and geometric methods. In this talk I will briefly review some of the main ideas in the proof and, if time permits and depending on the interest of the audience, I will try to highlight a few intriguing details that may be of independent interest.

We discuss some properties of spherical projections, and consider the related question regarding their congruency for two convex bodies. In particular, let $P$ and $Q$ be two convex polytopes both contained in the interior of an Euclidean ball $r\textbf{B}^{d}$. We prove that $P=Q$ provided that their spherical projections from any point on the sphere $rS^{d-1}$ are congruent. We also prove an analogous result for a pair of Euclidean balls.

We introduce floating bodies for convex, not necessarily bounded subsets of Rn. This allows us to define floating functions for convex and log concave functions and log concave measures. We establish the asymptotic behavior of the integral difference of a log concave function and its floating function. This gives rise to a new affine invariant which bears striking similarities to the Euclidean affine surface area. This is a joint work with Carsten Sch{\"u}tt and Elisabeth Werner.

Contact graphs have emerged as an important tool in the study of translative packings of convex bodies. The contact graph of a translative packing (that is, non-overlapping translates) of a convex body in Euclidean $d$-space is the (simple) graph whose vertices correspond to the packing elements with two vertices joined by an edge if and only if the two corresponding packing elements touch each other. The contact number of a finite translative packing of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. A translative packing of a convex body in Euclidean d-space is called a totally separable packing if any two packing elements can be separated by a hyperplane disjoint from the interior of every packing element. In this talk, we investigate the Hadwiger and contact numbers of totally separable translative packings of convex bodies.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.