Schedule for: 20w5037 - Geometric Tomography

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 Funk (Funk-Radon) transform, evaluating integrals of functions on the unit sphere in $\mathbb R^n$ over cross-sections by linear hyperspaces, is well studied. This transform has many applications, in medical imaging ($Q$-ball method in diffusion MRI), geometric tomography (intersection bodies problem). Last years, a similar transform (shifted Funk transform) associated with cross-sections of the unit sphere by $k$-planes passing through a fixed point (center), which is not necessarily the origin, appeared in the focus of researchers. The kernel of a shifted Funk transform with the center inside the unit sphere was described and inversion formulas were obtained. In my talk an universal approach will be discussed which allows to treat the case of arbitrarily located center. In most cases, shifted Funk transform has nontrivial kernel, so that single Funk data is not enough to recover a function on the unit sphere. Hence it is natural to ask when and how functions can be recovered from {\it multiple} Funk data. This question is completely answered for pairs of shifted Funk transforms. We fully describe all geometric configurations of the centers which provide injectivity of the paired transform and, correspondingly, unique recovery of functions on the unit sphere. A corresponding reconstruction procedure is given. The approach relies on the action of a hyperbolic automorphism group of the real ball and a billiard-like dynamics of a self-mapping of the unit sphere, generated by the set of centers. The common kernel of a pair of Funk transforms consists of related automorphic functions and is determined by the type of the above dynamics on the unit sphere.

A famous lemma in Newton's Principia says that the area of a segment of a bounded convex domain in the plane cannot depend algebraically on the parameters of the line that defines the segment. Vassiliev extended Newton's lemma to bounded convex domains in arbitrary even dimensions. In odd dimensions the volume cut out from an ellipsoid by a hyperplane depends not only algebraically but polynomially on the position of the hyperplane. Arnold conjectured in 1987 that ellipsoids in odd dimensions are the only cases in which the volume function in question is algebraic. The special case when the volume function is assumed to be polynomial was settled recently by Koldobsky, Merkurjev, and Yaskin. Motivated by a totally different problem I tried to construct a compactly supported distribution $f\ne 0$ whose Radon transform is supported in the set of tangent planes to the boundary surface $\partial D$ of a bounded convex domain $D \subset \mathbb R^n$. However, I found that such distributions can exist only if $\partial D$ is an ellipsoid. This result gives a new proof of the abovementioned special case of Arnold's conjecture.

Let $Q_n$ be the $n$-dimensional cube of side length one centered at the origin, and let $F$ be an affine $(n-d)$-dimensional subspace having distance to the origin less than or equal to $1/2$. We show that the $(n-d)$-dimensional volume of the section $Q_n$ by $F$ is bounded below by a value $c(d)$ depending only on the codimension $d$ but not on the ambient dimension $n$ or a particular subspace $F$. Joint work with Hermann Koenig.

We determine the non-central hyperplane sections of the $n$-simplex of maximal volume which have a fixed large distance to the centroid - large in the sense that the distance is bigger than the distance of the centroid to the midpoint if the edges. This complements similar results of Moody, Stone, Zach and Zvavitch for the $n$-cube and of Liu and Tkocz for the $n$-cross-polytope. We also show that parallels to the extremal hyperplanes for the $n$-simplex, the $n$-cube and the $n$-cross-polytope still provide at least local maxima for smaller distances, in a specified distance range and for sufficiently large dimensions (e.g. $n \ge 10$). Moreover, we find the maximal perimeters of non-central hyperplane sections of these bodies with large distances to the center. By perimeter we mean the $(n-2)$-dimensional intersection of the hyperplane with the boundary of the convex body.

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!

Let K be a convex body with volume one and barycentre at the origin. How small is the volume of the intersection of K and -K? I shall discuss such lower bounds and present applications to the Hadwidger covering/illumination conjecture. Based on joint work with H. Huang, B. Slomka and B. Vritsiou.

We shall discuss the Log-Brunn-Minkowski conjecture, a conjectured strengthening of the Brunn-Minkowski inequality proposed by Boroczky, Lutwak, Yang and Zhang. The discussion will involve introduction and explanation of how the local version of the conjecture arises naturally, a collection of ‘’hands on’’ examples and elementary geometric tricks leading to various related partial results, statements of related questions as well as a discussion of more technically involved approaches and results. Based on work with Johannes Hosle and Alexander Kolesnikov, as well as on previous joint results with Colesanti, Marsiglietti, Nayar, Zvavitch.

The conjectured log-Brunn-Minkowski inequality has attracted much interest since its introduction by Böröczky, Lutwak, Yang and Zhang in 2012. In this talk, I shall survey the connections between this inequality and various outstanding problems in convex geometry. Then, I shall discuss the recently discovered local version of the log-Brunn-Minkowski inequality and present a result of the author demonstrating the equivalence of this inequality to the original, 'global' log-Brunn-Minkowski inequality.

We discuss inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if $K, L$ are convex bodies in $\mathbb{R}^n$ with $|K|\theta^{\perp}| \le |L\cap \theta^{\perp}|$ for all $\theta \in S^{n-1}$, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John's position with $|K \cap \theta^{\perp}| \le |L|\theta^{\perp}|$ for all $\theta \in S^{n-1}$, then $|K| \le \sqrt{n}|L|$. We also discuss an extension of the result of Giannopoulos and Koldobsky to log-concave measures and an extension of the Loomis-Whitney inequality to positively concave and positively homogeneous measures.

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.

Our aim is to provide some estimates on the expected volume of various random convex sets, combining rearrangement inequalities and other tools from asymptotic geometric analysis: $\bullet$ For any ${\bf x}=(x_1,\ldots ,x_N)\in \oplus_{i=1}^N{\mathbb R}^n$ we denote by $T_{{\bf x}}=[x_1\cdots x_N]$ the $n\times N$ matrix with columns $x_1,\ldots ,x_N$. We discuss upper and lower bounds for the expected volume $${\mathbb E}_{\mu^N}\big({\rm vol}_n(T_{{\bf x}}(K))\big):= \int_{{\mathbb R}^n}\cdots \int_{{\mathbb R}^n}\big({\rm vol}_n(T_{{\bf x}}(K))\big)\,d\mu (x_N)\cdots d\mu (x_1)$$ of $T_{{\bf x}}(K)$, where $K$ is a centrally symmetric convex body in ${\mathbb R}^N$ and $\mu$ is an isotropic log-concave probability measure on ${\mathbb R}^n$. $\bullet$ Let $K$ be a centrally symmetric convex body in ${\mathbb R}^n$ and let $x_1,\ldots ,x_N$ be independent random points uniformly distributed in $K$. Given $r_1,\ldots ,r_N>0$, we discuss upper and lower bounds for the expected volume of the random polyhedron $\bigcap_{i=1}^NB(x_i,r_i)$. If time permits, we shall also discuss a question of V.~Milman asking for the equivalence of the norms $$\begin{equation*}\|{\bf t}\|_{K^s,K}=\int_{K}\cdots\int_{K}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_s\cdots dx_1\approx \|{\bf t}\|_2\end{equation*}$$ for all ${\bf t}\in {\mathbb R}^s$, where $K$ is a centrally symmetric convex body of volume $1$ in ${\mathbb R}^n$. The talk is based on joint works with G. Chasapis and N. Skarmogiannis.

The convex hull of $N$ independent random points chosen on the boundary of a simple polytope in $\mathbb R^n$ is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are derived. This is the first successful attempt of investigations which lead to rigorous results for random polytopes which are neither simple nor simplicial. The results contrast existing results when points are chosen in the interior of a convex set. Joint work with Matthias Reitzner and Elisabeth Werner.

In this talk I will discuss some properties of the spectrum of products of random matrices, like the rate of convergence on the triangular law and the asymptotic normality of the Lyapounov exponents. Based on a joint work with Boris Hanin.

We will discuss comparison inequalities between weak and strong moments of norms of random vectors with optimal (up to an universal factor) constants. We will also present applications to the concentration of log-concave random vectors and bounds on p-summing norms of finite rank operators. Based on a joint work with Piotr Nayar.

Let $M$ be a random $n\times n$ matrix with independent 0/1 random entries taking value 1 with probability $0 < p=p(n) < 1$. We provide sharp bounds on the probability that $M$ is singular for $C(\ln n)/n\leq p\leq c$, where $C, c$ are absolute positive constants. Roughly speaking, we show that this probability is essentially equal to the probability that $M$ has either zero row or zero column. Joint work with Konstantin Tikhomirov.

We extend the localization technique of Lovasz and Simonovits (1993) in the geometric form of Fradelizi and Guedon (2004) to the discrete setting. In particular, we show that the extreme points of the set of discrete log-concave random variables satisfying a linear constraint are log-affines with respect to a reference measure. Several applications are discussed akin to the continuous setting. This is joint with James Melbourne.

The Brunn-Minkowski and Prékopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of log-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa-Leindler inequality. Joint work with Jesus Rebollo Bueno.

The gaussian deviation inequalities occupy central role in the study of high-dimensional structures. For example, in the asymptotic theory of convex bodies, the distribution of a norm of the standard gaussian vector provides significant information about the geometry of lower-dimensional sections of the unit ball. It has been noticed, in several concrete situations, that lower deviation estimates exhibit stronger bounds than their upper analogues. In this talk, I will explain how we can integrate improved small deviation inequalities with tools drawn from the local theory of normed spaces to obtain optimal small-ball probabilities. Time permitting I will present applications of the latter to reverse Alexandrov inequalities. Based on a joint work with Grigoris Paouris and Konstantin Tikhomirov.

We study a variant of one of Lutwak's conjectures on the affine quermassintegrals of a convex body: Is it true that $$\frac{1}{\mathrm{vol}_n(K)^{\frac{1}{n}}} \left(\int_{G_{n,k}} \mathrm{vol}_k(P_F(K))^{-n}\,d\nu_{n,k}(F) \right)^{-\frac{1}{kn}} \leqslant c\sqrt{\frac{n}{k}}$$ holds for every convex body $K$ in $\mathbb{R}^n$ and all $1\leqslant k\leqslant n$, for some absolute constant $c>0$? Here integration is with respect to the rotation-invariant probability measure $\nu_{n,k}$ on the Grassmanian $G_{n,k}$ of all $k$-dimensional subspaces of $\mathbb{R}^n$, and $P_F$ denotes the orthogonal projection onto $F\in G_{n,k}$. We establish the validity of the above for a broad class of random polytopes in $\mathbb{R}^n$, that includes the case of random convex hulls with vertices chosen independently and uniformly from the interior or the surface of a convex body. We also discuss the case of unconditional convex bodies. Based on joint work with Nikos Skarmogiannis.

We will discuss a new proof of the classical Khintchine inequality for even moments, based on the concept of real rooted polynomials.

Given a convex body $K$ in $\mathbb R^n$ and a real number $p$, we study the extremal inner and outer affine surface areas $$IS_p(K) = \sup_{K'\subseteq K}\big(as_p(K')\big)$$ and $$os_p(K)=\inf_{K'\supseteq K}\big(as_p(K')\big),$$ where $as_p(K')$ denotes the $L_p$-affine surface area of $K'$, and the supremum is taken over all convex subsets of $K$ and the infimum over all convex compact subsets containing $K$. The convex body that realizes $IS_1(K)$ in dimension 2 was determined by B\'ar\'any. He also showed that this body is the limit shape of lattice polytopes in $K$. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate to give asymptotic estimates on the size of $IS_p(K)$. We use the L\"owner ellipsoid of $K$ to give asymptotic estimates on the size of $os_p(K)$. Based on joint work with Ohad Giladi, Han Huang and Carsten Schuett.

In 2013 Colesanti and Fragalà defined the surface area of a log-concave measure, and in 2015 Cordero-Erausquin and Klartag defined the moment measure of a convex function. These notions are basically the same. Comparing the results of the two papers raises some very natural questions about the surface area measure of log-concave functions. We will discuss the answers to a few of these questions.

The eighth Busemann-Petty problem asks the following question: If for an origin-symmetric convex body $K\subset{\mathbb R^n}$, $n \geq 3$, we have $$f_K(\theta)=C(vol_{n-1}(K\cap \theta^{\perp}))^{n+1}\qquad\forall \theta\in S^{n-1},$$ where the constant $C$ is independent of $\theta$, must $K$ be an ellipsoid? Here, $f_K$ is the is the curvature function (the reciprocal of the Gaussian curvature). We will show that the answer is affirmative for $K$ close enough to the Euclidean ball in the Banach-Mazur distance.

The classical Minkowski problem is a central problem in convex geometry which asks that given a nonzero finite Borel measure $\mu$, what are the necessary and sufficient conditions on $\mu$ such that $\mu$ equals to the surface area measure of a convex body $K$. In this talk, I will present the general dual extension of the classical Minkowski problem---the generalized dual Orlicz-Minkowski problem. That is, for which nonzero finite Borel measures $\mu$ on $\sphere$ and continuous functions $\wp$ and $\psi$ do there exist $\tau\in \R$ and $K\in \cK_{o}^n$ such that $\mu=\tau\,\deV(K,\cdot)$? Here $\deV(K,\cdot)$ is the finite Signed Borel measure. In particular, a solution where $G$ and $\psi$ are increasing to this problem will be presented. This work is based on a joint work with Professors Richard Gardner, Daniel Hug and Deping Ye.

We will explore old and new connections between optimal transport, cost-gradients and duality transforms on families of functions.

Given a finite metric space M, the set of Lipschitz functions on M with Lipschitz constant at most 1 can be identified with a convex body, say K(M), in R^n. The volume product P(M)=|K(M)| |K(M)º| is an isometric invariant of M. We discuss the extreme properties of the volume product. We show that if P(M) is maximal among all the metric spaces with the same number of points, then all triangle inequalities in M are strict and K(M)º is simplicial. We also focus on the metric spaces minimizing the volume product, and in the Mahler's conjecture for this class of convex bodies. In addition, we characterize the metric spaces such that K(M) is a Hanner polytope. This is a joint work with M. Alexander, M. Fradelizi, and A. Zvavitch.

The multiplicative structure on translation invariant valuations was introduced by the speaker about 15 years ago. Since than a number of non-trivial properties of this product has been discovered and several applications to integral geometry were found. In this talk we construct a complex analogue of the algebra of even translation invariant valuations and establish several properties of it. The construction and proofs use Radon and cosine transform on complex Grassmannians. If time permits we will describe briefly further analogues of this algebra over non-Archimedean local fields.

We present discrete analogs of classical volume inequalities for hyperplane sections. The main focus lies on a Meyer-type inequality for the lattice point enumerator, as it has been proposed by Gardner et al. We also discuss a reverse version of this inequality, where one replaces the coordinate hyperplanes by arbitrary lattice hyperplanes. This is joint work in progress with Ansgar Freyer.

We will discuss a collection of relatively recent results in harmonic analysis on the discrete cube, involving concepts such as influences, tail spaces, functions of bounded degree and heat semigroup.

The extremal sets in the Alexandrov-Fenchel inequality are by and large widely open, even in the case where all the bodies are polytopes. I will discuss ongoing work with Ramon van Handel on the polytope extremals.

Assume that Earth is made out of a transparent glass and contains a convex body $K$ in its interior. Let $K$ be seen as a disk from every point on the planet's surface, possibly of different radii. Can one conclude that $K$ is a Euclidean ball? What if it is seen as an ellipse or a polygon? We discuss related open problems, provide known and recent results that answer all of the questions above, as well as their dual counterparts for non-central sections of convex bodies.

In convex geometry and geometric tomography an important reverse form of the Brunn-Minkowski inequality is Rogers-Shephard inequality, which reads: for any convex body $K \subset \R^n$: $$\vol_n(K-K) \leq \binom{2n}{n} \vol_n(K),$$ where $K-K = \{x-y \colon x,y \in K\}$. Another inequality due to Rogers and Shephard is the so-called projection-section inequality: given any convex body $K \subset \R^n$ containing the origin, and any $m$-dimensional subspace $H$ of $\R^n$, $$\vol_m(K \cap H) \vol_{n-m}(P_{H^{\perp}}(K)) \leq \binom{n}{i} \vol_n(K),$$ where $P_H(K)$ denotes the orthogonal projection of $K$ onto complement $H^{\perp}$ of $H$. In this talk we address functional forms of the above inequalities for functions having certain concavity conditions.

We give a new proof of Fejes Tóth’s zone conjecture: for any sequence $v_1,v_2,...,v_n$ of unit vectors in a real Hilbert space $H$, there exists a unit vector $v$ in $H$ such that $$\begin{equation*} |\langle v_k,v \rangle| \geq \sin(\pi/2n) \end{equation*}$$ for all $k$. This can be seen as sharp version of the plank theorem for real Hilbert spaces. As a result, we obtain a unified approach to some of the most important plank problems on the real and complex setting: the classic plank problem, the complex plank problem, Fejes Tóth’s zone conjecture and the strong polarization problem.

I'll present a result of Assaf Naor, Gilles Pisier and myself: Let $S_1$ denote the Schatten--von Neumann trace class, i.e., the Banach space of all compact operators $T:\ell_2\to \ell_2$ whose trace class norm $\|T\|_{S_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$. We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $S\subset S_1$ with $|S|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $S_1$. This extends a well known result of Brikmann and Charikar (2003) who proved a similar result with $\ell_1$ replacing $S_1$. It stand in sharp dichotomy with the Johnson--Lindenstrauss lemma (1984) which says that the situation in $\ell_2$ is very different.

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.