Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls

Radosław Adamczak^{1}, Peter Pivovarov^{2}, Paul Simanjuntak^{2}

^{1}University of Warsaw, Poland; ^{2}University of Missouri, USA

I will present central limit theorems for the volumes of random sections of fixed codimension of high dimensional $\ell_p^n$-balls. I will also discuss some related open questions.

The intrinsic volume metric and new upper bounds for optimal polytopal approximation

Florian Besau^{1}, Steven Hoehner^{2}

^{1}Technische Universität Wien, Austria; ^{2}Longwood University, U.S.A.

We introduce a new notion of metric for the class of convex bodies in $d$-dimensional Euclidean space, measuring distance based on the intrinsic volume difference (or quermassintegral difference). This novel metric serves as a basis for establishing improved upper bounds for the asymptotic optimal approximation of the Euclidean unit ball by a polytope in general position.
Our results demonstrate an improvement by a factor of the dimension when compared to classical results on the asymptotic approximation of the intrinsic volume difference with polytopes restricted to the domain of the ball.
In our proof, we rely on a probabilistic construction to bound the asymptotic optimal polytope by random polytopes and utilize integral formulas by Blaschke--Petkantschin and Chern.

High-dimensional asymptotics of the lr Grothendieck problem and r to p norms of Levy matrices

Kavita Ramanan, Xiaoyu Xie

Brown University, United States of America

Given an n × n symmetric matrix A, the lr Grothendieck problem considers the maximum of the quadratic form x’ A x over vectors x in the unit lr sphere. Both this quantity and the r to p operator norm of the matrix A arise in a variety of fields including asymptotic convex geometry, theoretical computer science and statistical mechanics, and are known to be hard to compute or even efficiently approximate in various parameter regimes. In this talk, we identify the high-dimensional asymptotics of these quantities (suitably renormalized) for symmetric random matrices with independent and identically distributed heavy-tailed upper-triangular entries whose index lies in a suitable parameter regime. This complements results in the well studied spectral case when both r and p are equal to 2, as well as asymptotics for operator norms of Gaussian and non-negative random matrices. The analysis uses different techniques that are not reliant on spectral methods, and yields new scaling limits that do not arise in the spectral setting. As a corollary of our results we also characterize the limiting ground state of the Levy spin glass model when the heavy-tail index lies between 0 and 1. This is joint work with Xiaoyu Xie.