Schedule for: 23w5053 - Harmonic Analysis and Convexity

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

Lounge above the reception

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.

We determine the maximal non-central hyperplane sections of the $l_1^n$-ball if the fixed distance of the hyperplane to the origin is between $\frac 1 {\sqrt 3}$ and $\frac 1 {\sqrt 2}$. This adds to a result of Liu and Tkocz who considered the distance range between $\frac 1 {\sqrt 2}$ and $1$. For $n \ge 4$, the maximal sections are parallel to the $(n-1)$-dimensional coordinate planes. We also show that the maximal non-central hyperplane sections of the regular $n$-simplex of side-length $\sqrt 2$ at a fixed distance $t$ to the centroid are those parallel to a face of the simplex, if $\sqrt{\frac {n-2} {3 (n+1)}} < t \le  \sqrt{\frac n { n+1}}$ and $n \ge 5$. For $n=4$, the same is true in a slightly smaller range for $t$.  We also study non-central sections of the complex $l_1^2$- and $l_\infty^2$-balls, where the formulas are more complicated than in the real case. Also, the extrema are partially different than in the real case.

We are computing the Banach-Mazur distances between the spaces $\ell_{p}^{n}\otimes_{\pi}\ell_{q}^{n}$, $1\leq p,q\leq\infty$ and the spaces $\ell_{1}^{n^{2}}$ and $\ell_{\infty}^{n^{2}}$.

I shall present an extension of Szarek’s optimal Khinchin inequality (1976) for distributions close to the Rademacher one when all the weights are uniformly bounded by a 1/√2 fraction of their total l_2 mass; similarly for Ball’s cube slicing inequality (1986). The underpinning to such estimates is the Fourier-analytic approach going back to Haagerup (1981). Based on joint work with Eskenazis and Nayar.

The surface area of convex bodies is monotonic with respect to set-inclusion. Also, If A,B and C are convex bodies, then the Lebesgue measure satisfies the following supermodularity inequality for their Minkowski sums: |A+B| + |A+C| < |A| + |A+B+C|. In this talk, based on a joint work with M. Fradelizi, M. Madiman and A. Zvavitch, we explore weighted analogues of these properties by replacing the Lebesgue measure with a Radon measure. We verify that a Radon measure with the supermodularity property must be the Lebesgue measure. We then consider restricted versions of the problem and its relation to the monotonicity of weighted surface area.

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 PDC front desk 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 will outline the proof that an origin-symmetric star-convex body $K$ with sufficiently smooth boundary and such that every hyperplane section of $K$ passing through the origin is a body of affine revolution, is itself a body of affine revolution. This will give a positive answer to the recent question asked by G. Bor, L. Hern\'andez-Lamoneda, V. Jim\'enez de Santiago, and L. Montejano-Peimbert [2, Remark 2.9], though with slightly different prerequisites. Our argument is built mainly upon the tools of differential geometry and linear algebra, but occasionally we will need to use some more involved facts from other fields like algebraic topology or commutative algebra. The talk is based on the article [2]. $$\,$$ [1] G. Bor, L. Hern ́andez-Lamoneda, V. Jim ́enez de Santiago, and L. Montejano, On the isometric conjecture of Banach, Geometry & Topology 25 (2021), no. 5, 2621–2642. $$\,$$ [2] B. Zawalski, On star-convex bodies with rotationally invariant sections, Beitr ̈age zur Algebra und Geometrie (2023).

The vector balancing constant of two symmetric convex bodies $ U, V \subset \mathbb R^n$, $ \beta(U, V)$, is defined to be the smallest $ b > 0 $ such that for any $n$ vectors in $U$, there exists a signed combination of the vectors which lies in $bV$. Komlós conjecture asks whether $\beta(B_2^n, B_\infty^n)$ is bounded by a universal constant, where $B_p^n$ denotes the $l_p$-ball, in dimension $n$. We will introduce a related parameter $\alpha(U, V) \leq \beta(U,V)$, defined via lattices, and review some known inequalities for these two parameters. In 1997, W. Banaszczyk and S. Szarek showed that if a convex body has gaussian measure $\gamma_n(V)\geq 1/2$, then $\alpha(B_2^n, V) \leq c$ (for some universal $c>0$); this yields $\alpha(B_2^n, B_\infty^n)\leq c \sqrt{2 \log n}$ for the cube. They conjecture that a similar inequality holds for convex bodies of gaussian measure $p<1/2$, i.e., that there exists a non-increasing function $f$ (independent of $n$) such that $\beta(B_2^n, V)\leq f(\gamma_n(V))$. We answer this question in the affirmative for the parameter $\alpha$.

We present a short version of the original proof of Ball's complex plank theorem.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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 will discuss a problem concerning random frames which arises in signal processing. A frame is an overcomplete set of vectors in the n-dimensional linear space which allows a robust decomposition of any vector in this space as a linear combination of these vectors. Random frames are used in signal processing as a means of encoding since the loss of a fraction of coordinates does not prevent the recovery. We will discuss a question when a random frame contains a copy of a nice (almost orthogonal) basis. Despite the probabilistic nature of this problem it reduces to a completely deterministic question of existence of approximately Hadamard matrices. An n by n matrix with plus-minus 1 entries is called Hadamard if it acts on the space as a scaled isometry. Such matrices exist in some, but not in all dimensions. Nevertheless, we will construct plus-minus 1 matrices of every size which act as approximate scaled isometries. This construction will bring us back to probability as we will have to combine number-theoretic and probabilistic methods. $$\,$$ Joint work with Xiaoyu Dong.

We show that for log-concave real random variables with fixed variance the Shannon differential entropy is minimized for an exponential random variable. Based on a joint work with James Melbourne and Cyril Roberto.

The classical theory of random polytopes addresses questions such as computing the expectation or variance of geometric parameters associated to a random polytope (e.g., volume, number of facets, or mean width); more recent theory also aims to obtain concentration of measure for such quantities. The new theory of higher-order projection bodies naturally leads to a question in random polytopes which current theory, surprisingly, does not address: bounding a high negative moment of the mean width of a certain random polytope, which requires bounding the probability that this mean width is a small fraction of its expectation ("small-ball estimates"). These small-ball estimates use different tools from those commonly employed in the field of random polytopes, and it turns out that the behavior of the negative moment demonstrates a phase transition. We will conclude by mentioning some related open problems. $$\,$$ Joint with J. Haddad, D. Langharst, M. Roysdon, and D. Ye.

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

In this talk we consider the stability and equality case in the Gaussian B-inequality of Cordero-Erausquin, Fradelizi, and Maurey. The talk is based on a joint work with Galyna Livshyts, Liran Rotem, and Alexander Volberg.

Inspired by the Busemann-Petty problem in Convex Geometry, we examine similar tomography questions for even, continuous functions concerning estimates for their L_p-norms given information from Radon-type transforms of the functions. In particular, we will study comparison problems for the spherical and classical Radon transforms by introducing families of functions which extend the class of intersection bodies of star bodies due to Lutwak. If time permits, we will also discuss comparison problems for the (n-k)-dimensional Radon and spherical Radon transforms. $$\,$$ Based on a joint work with Alexander Koldobsky and Artem Zvavitch.

Suppose we wanted to illuminate a solid object with convex shape, that is, illuminate its surface, by placing a number of light sources around it. What is the smallest number of light sources we would need? This seemingly innocent question has actually turned into a longstanding conjecture in Convex and Discrete Geometry, called the Illumination Conjecture. The conjecture states that for an $n$-dimensional object, we should need less than $ 2^n $ light sources, except if the object ``looks like'' a cube (which then needs $ 2^n $). In my talk, I will start with some pictures, then speak about historical progress on the conjecture. I will finish with presenting some recent results of mine with my supervisor on convex objects with a lot of symmetries.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

I will present some concentration inequalities on the Unitariy group and some applications. Based on Joint work with Kavita Ramanan.

We show (via 3 different methods) that the first Dirichlet eigenfunction of the Ornstein-Uhlenbeck operator on a symmetric convex body is log-concave. As an application, we get a version of the Brunn-Minkowski inequality for the first Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator (aka the Gaussian principal frequency). In the Lebesgue case, these results are classical, but nothing of this sort was known for other log-concave measures. Curiously, all our methods are rather limited to the Gaussian case. Joint work with Colesanti, Francini and Salani.

We prove a Rogers-Brascamp-Lieb-Luttinger inequality for functions defined in the space of $n \times m$ matrices, using a particular form of fiber-symmetrization. Some applications on symmetrization of matrix norms are given. We also discuss a conjectured inequality by Schneider, on the higher-order difference body.

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

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies in the symmetric difference metric, or other metrics coming from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance. For a polygon Q in the plane, the convex hull of n points chosen at random on the boundary of Q gives a random polygon Q_n. We determine the exact limiting behavior of the expected Hausdorff distance between Q and a random polygon Q_n as the number n of points chosen on the boundary of Q goes to infinity. Based on joint work with J. Prochno, C. Schuett and M. Sonnleitner.

Borsuk’s number f(n) is the smallest integer such that any set of diameter 1 in n-dimensional Euclidean space can be covered by f(n) sets of a smaller diameter. Exponential upper bounds on f(n) were first obtained by Schramm (1988) and later by Bourgain and Lindenstrauss (1989), while a lower bound (exponential in n^{1/2}) was obtained by Kahn and Kalai (1993). $$\,$$ To obtain an upper bound on f(n), Bourgain and Lindenstrauss provided exponential bounds (both upper and lower) in Grünbaum's problem – the problem of determining the minimal number of open balls of diameter 1 needed to cover a set of diameter 1. $$\,$$ On the other hand, Schramm provided an exponential upper bound on the illumination number of n-dimensional bodies of constant width. Kalai (2015) asked for a corresponding lower bound, namely if there exists an n-dimensional convex body of constant width with the illumination number exponential in n. $$\,$$ In this talk I will outline the construction that answers Kalai’s question in the affirmative and provide a new lower bound in the Grünbaum’s problem. The talk is based on a joint work with Andriy Bondarenko and Andriy Prymak.

The Minkowski type problems for convex bodies are central in convex geometry with applications in analysis, affine geometry, partial differential equations, etc. Such problems aim to solve measure equations and hence provide characterizations on geometric measures derived from the variational formulas of geometric invariants in terms of perturbations of convex bodies. $$\,$$ In this talk, I will talk about the Minkowski problem for certain unbounded convex sets called $C$-compatible sets. Our focus is on the dual Minkowski problem with the geometric measure produced by the first order derivative of the $q$-th dual volume of $C$-compatible sets in terms of the logarithmic addition.

Suppose Alice wants to communicate with Bob using a collection of points $K$ in space. However, the night is foggy, so Bob receives the random point $x + W$ when Alice sends $x$, where $W$ is uniformly distributed on the unit ball. Does communication suffer if the points in $K$ are brought pairwise closer together? Using methods of rearrangement and majorization, in a joint project with G. Aishwarya, I. Alam, D. Li and O. Zatarain-Vera, we affirmatively answer this Information-theoretic question in various cases. By-products of our work describe the natural behavior of intrinsic volumes of convex bodies under contractions.

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

It has been observed, by Froissart (1969), that zeros and poles of high order Padé approximants of random perturbations of a deterministic Taylor series tend to form unstable pairs. These pairs appear at loci characteristic of the random part in the coefficients of the Taylor series. While this phenomenon has only been confirmed experimentally, it has been suggested, and indeed broadly used, as a noise detection tool. In this talk we will explain how one can combine methods from high-dimensional probability and logarithmic potential theory to rigorously establish and quantify this phenomenon for the ``pure noise’’ case, when the coefficients come from some distribution with anti-concentration properties. Based on a joint ongoing work with S. Dostoglou (University of Missouri).

We establish Central Limit Theorem (CLT) for the volumes of intersections of $ B_p^n $, $ 0 < p < 2 $, with uniform random subspaces of fixed codimension d as n tends to infinity. This result is obtained using volume representation as sum of Gaussian mixtures. As a corollary we obtain higher order approximations for expected volumes, refining previous results by Koldobsky and Lifshits and approximation obtained from the Eldan-Klartag version of CLT for convex bodies. Based on a work with R. Adamczak and P. Pivovarov.

Characterizing the Euclidean space among all normed spaces is one of the aims of the ten problems formulated in 1956 by Busemann and Petty. These problems lead to the study of certain integral operators on convex bodies, such as the intersection body operator for the first Busemann-Petty problem. In the class of convex bodies, obtaining global statements about the fixed points of such operators is difficult. The local study of these problems appears to be a more approachable initial step, which has yielded local solutions to problems 5 and 8 by Alfonseca, Nazarov, Ryabogin and Yaskin. We will apply similar techniques to study the fixed points up to dilation of the $p$-centroid body operator in a neighborhood of the Euclidean ball. Given an origin-symmetric convex body $K$, this body can be identified by its radial function $\rho_K$ or its support function $h_K$ where \[\rho_K(\theta):=\max\{r\hspace{3pt}\mid\hspace{3pt} r\theta\in K\}\hspace{30pt}\text{ and }\hspace{30pt} h_K(\theta):=\max\{\theta\cdot x\hspace{3pt}\mid\hspace{3pt} x\in K\}\] for all $\theta\in{\mathcal{S}^{n-1}}$. For $p\geq 1$, the $p$-centroid body of $K$ is denoted $\Gamma_p K$and may be defined using the radial and support function by \[h_{\Gamma_p K}(\theta):=\frac1{\text{vol}_n(K)}\int_K\left\lvert\theta\cdot x\right\rvert^p\,dx,\hspace{40pt}\forall \theta\in{\mathcal{S}^{n-1}}\] $$\,$$ When $p=1$, the boundary of $\Gamma_pK$ can be described physically: If $K$ has density $1/2$ and were allowed to float in a particular orientation, then the boundary of $\Gamma_p$ is the locus of the center of mass of the submerged portion for all orientations. In the integral definition, passing to polar coordinates yields $h_{\Gamma_pK}=c_K\,\mathcal{C}\rho_K^{n+p}$ where $c_K$ is some constant depending on the body and $\mathcal{C}$ is the cosine transform. Since the eigenspaces of the cosine transform are the spherical harmonics, we will use techniques from harmonic analysis to show that if $K$ is close to the Euclidean ball and $K=c\Gamma_pK$ for some real number $c$, then $K$ is the Euclidean ball up to linear transformation.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.