Schedule for: 17w5074 - Recent Advances in Discrete and Analytic Aspects of Convexity

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.

We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs, and for the graph of the packing, there is only one equilibrium stress up to scaling. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. Joint work with Evan Solomonides and Maria Yampolskaya.

The Kneser--Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We prove the Kneser-Poulsen conjecture for uniform contractions whenever N is sufficiently large (depending only on d) in Euclidean, spherical as well as hyperbolic d-space for all d>1.

The Kneser--Poulsen Conjecture states that if the centers of a family of $N$ unit balls in ${\mathbb E}^d$ is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). A 'uniform contraction' is a contraction where all the pairwise distances in the first set of points are larger than all the pairwise distances in the second set of points. We show that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $N\geq(1+\sqrt{2})^d$. Joint work with K\'aroly Bezdek.

The Kneser-Poulsen conjecture says that if a finite set of (not necessarily congruent) balls in an n-dimensional Euclidean space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of the balls does not increase as well. We give new results about central sets of subsets of a Riemannian manifold and apply these results to prove new special cases of the Kneser-Poulsen conjecture in the two-dimensional sphere and the hyperbolic plane.

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!

I will present extensions of classical results on affine invariance due to Grinberg and their corresponding inequalities due to Busemann-Strauss and Grinberg to the setting of flag manifolds. I will also present some inequalities for the dual quantities in the sett of convex bodies as well as extensions in the set of functions. The talk will be based on a joint work with S. Dann and P. Pivovarov.

Alexandrov's inequalities imply that for any convex body A, the sequence of intrinsic volumes is non-increasing (when suitably normalized). Milman's random version of Dvoretzky's theorem shows that a large initial segment of this sequence is essentially constant, up to a critical parameter called the Dvoretzky number. We show that this near-constant behavior actually extends further, up to a different parameter associated with A. This yields a new quantitative reverse inequality that sits between the approximate reverse Urysohn inequality, due to Figiel--Tomczak-Jaegermann and Pisier, and the sharp reverse Urysohn inequality for zonoids, due to Hug--Schneider. In fact, we study concentration properties of the volume radius and mean width of random projections of A and show how these lead naturally to such reversals. Joint work with Grigoris Paouris and Petros Valettas.

The concentration of measure is perhaps one of the most important tools in modern probability with profound impact in asymptotic convex geometry. In Gauss' space states that any $L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies: $$\gamma_n(\{ x: |f(x) - M|>t\})\leq \exp(-t^2/(2L^2)),$$ for all $t>0$, where $M$ is a median of $f$. A consequence of this inequality is that ${\rm Var}(f) \leq L^2$. However, there are many Lipschitz maps for which ${\rm Var}(f) \ll L^2$. In light of that, one may ask if we could replace the Lipschitz constant in the above the concentration inequality by the variance. Although this is not correct in general, we will see that for convex maps a variance-sensitive, one-sided small deviation inequality can always be obtained. Based on joint work with G. Paouris (Texas A & M).

Given a convex body $K$ and a real number $a$, how should we define the convex body $K^a$? One way to answer this question is to write down a list of natural properties we expect from such a power, and then prove existence and uniqueness of a construction satisfying these properties. In this talk we will explain why a natural power operation does not exist for $a > 1$, but does exist for $0 < a < 1$. We will also discuss the uniqueness question, which is more delicate. Based on joint work with Vitali Milman.

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.

The first recent result, obtained jointly with J. Grote, generalizes a theorem by Ludwig, Schuett and Werner on approximation of a convex body K in the symmetric difference metric by an arbitrarily placed polytope with a fixed number of vertices. The second recent result is by S.Hoehner, C. Schuett and E. Werner. It gives a lower bound, in the surface deviation, on the approximation of the Euclidean ball by an arbitrary positioned polytope with a fixed number of k-dimensional faces.

We prove a new family of Prekopa-Leindler type inequalities corresponding (in the case of geometric convex functions) to the linear structure induced by the geometric duality A.

We will discuss connections between Gromov's work on isoperimetry of waists and Milman's work on the $M$-ellipsoid of a convex body. It is proven that any convex body $K \subseteq \mathbb{R}^n$ has a linear image $\tilde{K} \subseteq \mathbb{R} ^n$ of volume one satisfying the following waist inequality: Any continuous map $f:\tilde{K} \rightarrow \mathbb{R}^{\ell}$ has a fiber $f^{-1}(t)$ whose $(n-\ell)$-dimensional volume is at least $c^{n-\ell}$, where $c > 0$ is a universal constant. Already in the case where f is linear, this constitutes a slight improvement over known results. In the specific case where $K = [0,1]^n$, one may take $\tilde{K} = K$ and $c = 1$, confirming a conjecture by Guth. We furthermore exhibit relations between waist inequalities and various geometric characteristics of the convex body $K$.

Gromov and Memarian (2003--2011) have established the \emph{waist inequality} asserting that for any continuous map $f : \mathbb S^n \to \mathbb R^{n-k}$ there exists a fiber $f^{-1}(y)$ such that every its $t$-neighborhood has measure at least the measure of the $t$-neighborhood of an equatorial subsphere $\mathbb S^{k}\subset \mathbb S^n$. Going to the limit we may say that the $(n-k)$-volume of the fiber $f^{-1}(y)$ is at least that of the standard sphere $\mathbb S^{k}$. We extend this limit statement to the exact bounds for balls in spaces of constant curvature, tori, parallelepipeds, projective spaces and other metric spaces. By the volume of preimages for a non-regular map $f$ we mean its \emph{lower Minkowski content}, some new properties of which will be also presented in the talk. Joint work with Roman Karasev and Alfredo Hubard.

The {\em disjointness graph} $G=G({\cal S})$ of a set of segments ${\cal S}$ in ${R}^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of $G$ is bounded by a function of $\omega(G)$, the clique number of $G$. More precisely, we have $\chi(G)\le(\omega(G))^4+(\omega(G))^3$. It follows, that $\cal S$ has $\Omega(n^{1/5})$ pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing $\omega(G)$ and $\chi(G)$ for disjointness graphs of lines in space are NP-complete tasks. However, we can design efficient algorithms to compute proper colorings of $G$ in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free ($\omega(G)=2$), but whose chromatic numbers are arbitrarily large. Joint work with G\'abor Tardos and G\'eza T\'oth.

Any graph G can be embedded in a Euclidean space as a contact graph of sphere packing. In this talk we consider contact graphs of packings by congruent spheres in Euclidean and spherical spaces. In particular, we compute explicitly the minimal dimensions of representations for the join of graphs. We also show that analogs of Steiner's porism and Soddy's hexlet in higher dimensions can be found via packings by congruent spheres of spherical spaces.

We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from \cite{R2}. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \cite{R} and \cite{R2} are answered, joint work with Sergii Myroshnychenko and Dmitry Ryabogin.

Gardner and Golubyatnikov asked whether two continuous functions on the sphere coincide up to reflection in the origin if their restrictions to any great circle coincide after some rotation. In this talk we will discuss two modifications of this problem. Let $K$ and $L$ be convex bodies in $\mathbb R^3$ such that their sections by cones or non-central planes are directly congruent. We will show that if their boundaries are of class $C^2$, then $K$ and $L$ coincide.

Lattice polytopes are convex hulls of finitely many points with integer coordinates in ${\mathbb R}^n$. A function $Z$ from a family ${\cal F}$ of subsets of ${\mathbb R}^n$ with values in an abelian group is a valuation if $$Z(P)+Z(Q)=Z(P\cup Q)+Z(P\cap Q)$$ whenever $P,Q,P\cup Q,P\cap Q\in{\cal F}$ and $Z(\emptyset)=0$. The classification of real-valued invariant valuations on lattice polytopes by Betke \& Kneser is classical (and will be recalled). It establishes a characterization of the coefficients of the Ehrhart polynomial. Building on this, classification results are established for Minkowski and tensor valuations on lattice polytopes. The most important tensor valuations are the discrete moment tensor of rank $r$, $$ L^r(P)=\frac1{r!}\sum_{x\in P\cap{\mathbb Z}^n}x^r, $$ where $x^r$ denotes the $r$-fold symmetric tensor product of the integer point $x\in{\mathbb R}^n$, and its coefficients in the Ehrhart polynomial, called Ehrhart tensors. However, there are additional examples for tensors of rank nine with the same covariance properties. (Based on joint work with K\'aroly J. B\"or\"ozcky and Laura Silverstein)

To this day, the starting point for many important developments in valuation theory is Hadwiger's remarkable characterization of continuous rigid motion invariant real valued valuations as linear combinations of the intrinsic volumes. A fascinating open problem, dating back to work of R. Schneider from 1974, is the characterization of continuous {\it convex body valued} valuations which intertwine rigid motions. In this talk, we present some of the recent progress regarding this question. In particular, we discuss three different ways to represent an even such valuation, namely via Crofton measures, Klain bodies, and generating functions.

The notion of mixed volumes $V(K_1,\dots, K_d)$ of convex bodies $K_1,\dots ,K_d$ in Euclidean space $\rd$ is of central importance in the Brunn-Minkowski theory. Representations for mixed volumes are available in special cases, for example as integrals over the unit sphere with respect to mixed area measures. More generally, in Hug-Rataj-Weil (2013) a formula for $V(K [n], M[d-n]), n\in \{1,\dots ,d-1\},$ as a double integral over flag manifolds was established which involved certain flag measures of the convex bodies $K$ and $M$ (and required a general position of the bodies). In the talk, we discuss the general case $V(K_1[n_1],\dots , K_k[n_k]), n_1+\cdots +n_k=d,$ and show a corresponding result involving the flag measures $\Omega_{n_1}(K_1;\cdot),\dots, \Omega_{n_k}(K_k;\cdot)$. For this purpose, we first establish a curvature representation of mixed volumes over the normal bundles of the bodies involved. We also point out a connection of the latter result to a combinatorial formula of R. Schneider, in the case of polytopes. Joint work with Daniel Hug (Karlsruhe) and Jan Rataj (Prague).

Recently Huang, Lutwak, Yang and Zhang introduced a broad class of geometric measures related to convex bodies. Among these are the dual curvature measures ${\widetilde C}_q(K,\cdot)$ of an $n$-dimensional convex body $K$ and $q\in\R$. They are the counterparts to the classical curvature measures of convex bodies in the dual Brunn-Minkowski theory. Here we discuss the associated dual Minkowski problem and present tight subspace concentration inequalities for the dual curvature measures of origin-symmetric convex body for q ≥ n + 1. This supplements former results obtained in the range q ≤ n. Based on a joint work with Hannes Pollehn.

I'll present a few inequalities on metric spaces holding for $L_p$ and other natural spaces. One of these inequalities can serve as the metric analogue of (Pisier's) property $\alpha$ and serves as an obstruction to the Lipschitz(and uniform) embeddability of (some discrete subsets of) Schatten classes into $L_p$ spaces. Joint work with Assaf Naor.

we study the volume ratio of the projective tensor products $\ell^n_p\otimes_{\pi}\ell_q^n\otimes_{\pi}\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. The asymptotic formulas we obtain are sharp in almost all cases. As a consequence of our estimates, these spaces allow for an almost Euclidean decomposition of Kashin type whenever $1\leq p \leq q\leq r \leq 2$ or $1\leq p \leq 2 \leq r \leq \infty$ and $q=2$. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype $2$ constant, we obtain information on the cotype of these $3$-fold projective tensor products. Our results naturally generalize to the $k$-fold products $\ell_{p_1}^n\otimes_{\pi}\dots \otimes_{\pi}\ell_{p_k}^n$ with $k\in\N$ and $1\leq p_1 \leq \dots\leq p_k \leq \infty$. Joint work with Giladi, Prochno, Tomczak-Jaegermann and Werner.

Let B be an unconditional convex body in R^n in the ell-position. Then for any small epsilon, and for random subspace E of dimension epsilon*log(n)/log(1/epsilon), distributed according to the rotation-invariant (Haar) measure, the section of B cut by E is (1+C*epsilon)-Euclidean with probability close to one. This shows that the ``worst-case'' dependence on epsilon in the randomized Dvoretzky theorem in the ell-position is significantly better than in John's position.

Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\R^n$. We show that the random vector $Y=T(X)$ satisfies $$ \mathbb{E} \sum \limits_{j=1}^k j\mobx{-}\min _{i\leq n}{X_{i}}^2 \leq C \mathbb{E} \sum\limits_{j=1}^k j\mobx{-}\min _{i\leq n}{Y_{i}}^2 $$ for all $k\leq n$, where ``$\jm$'' denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen--Lo\`eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian) which are of independent interest. This is a joint work with Konstantin Tikhomirov.

Let $K$ be a convex body in $\mathbb{R}^n$ whose centroid is at the origin, let $E\in G(n,k)$ be a subspace, and let $\xi\in S^{n-1}$. I will discuss my joint work with Ning Zhang, where we found the best constant $c = \left(\frac{k}{n+1}\right)^k$ so that $|(K|E)\cap\xi^+|_k\geq c\,|K|E|_k$. Here, $|\cdot|_k$ is $k$-dimensional volume, $K|E$ is the projection of $K$ onto $E$, and $\xi^+ = \{ x\in\mathbb{R}^n:\, \langle x,\xi\rangle\geq 0\}$. Our result generalizes both Gr\"unbaum's inequality, and an old inequality of Minkowski and Radon.

A flag area measure on a finite-dimensional euclidean vector space is a continuous translation invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p + 1)$-dimensional linear subspace containing $v$. Using local parallel sets, Hinderer constructed examples of $SO(n)$- covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general space of $SO(n)$-covariant flag area measures, which satisfy a similar formula for polytopes, but with an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Hinderer's flag area measure correspond to the special case where the elementary symmetric polynomial is just the product. We also provide a classification result in the spirit of Hadwiger's theorem. We introduce a natural notion of smoothness and show that every smooth $SO(n)$-covariant flag area measure is a linear combination of the ones which we constructed. Joint work with Judit Abardia-Ev\'equoz and Andreas Bernig.

We discuss a construction of convex bodies from non-degenerate Borel measures. Using these bodies, we consider variants of problems of approximating convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we provide a lower bound for certain average norms of centroid bodies of non-degenerate probability measures. Based on joint work with Han Huang.

We will discuss the maximal values 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 bounded number of vertices where $K^z = \{y\in{\mathbb R}^n : (y-z) \cdot(x-z)\le 1, \mbox{\ for all\ } x\in K\}$ is the polar body of $K$ with respect to the center of polarity $z$. In particular, we will discuss polytopes with $n+2$ vertices in $\RR^n$, symmetric polytopes with $8$ vertices in $\mathbb{R}^3$, and that the supremum is reached at a simplicial polytope with exactly $m$ vertices for all convex bodies of $m$ or fewer vertices.

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.