(22 mars 2024/March 22, 2024) Colloque des sciences mathématiques du Québec/CSMQ.
https://www.crmath.ca/en/activities/#...

Peter Pivovarov: How to prove isoperimetric inequalities by random sampling

Abstract: In the class of convex sets, the isoperimetric inequality can be derived from several different affine inequalities. Fundamental constructions of convex sets, such as polar bodies and centroid bodies, all satisfy strengthened isoperimetric theorems, as proved by Blaschke, Busemann and Petty. A powerful analytic framework for kindred problems was developed by Lutwak, Yang and Zhang, with their introduction of Lp affine isoperimetric inequalities. Establishing isoperimetric inequalities for the highly non-convex Lp objects when p less than 1 (or p is even negative), has proved to be a challenge due to the lack of convexity. However, this range of p is important to bridge inequalities between Brunn-Minkowski theory and dual Brunn-Minkowski theory. I will discuss a probabilistic approach to proving Lp affine isoperimetric inequalities in the non-convex range. Gems from geometric probability, going back to Sylvester's famous four point problem, motivate empirical definitions of polar bodies and their Lp-analogues. These empirical versions turn out to be more susceptible to convex analytic methods, and in turn provide a bridge between the convex and non-convex worlds. Based on joint work with R. Adamczak, G. Paouris, and P. Simanjuntak.

Supported by NSF Grant DMS-2105468