Randomized isoperimetric inequalities

TL;DR

This paper explores randomized versions of classical isoperimetric inequalities for convex sets, demonstrating stochastic dominance in various geometric functionals and their implications in convex geometry and probability.

Contribution

It introduces stronger randomized forms of classical inequalities, showing how stochastic dominance can recover traditional results and extend their applications.

Findings

Randomized inequalities exhibit stochastic dominance for volume, surface area, and mean width.

Randomized forms recover classical inequalities via laws of large numbers.

Applications include advances in convex geometry and probabilistic analysis.

Abstract

We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities admit stronger randomized forms in the following sense: for natural families of associated random convex sets one has stochastic dominance for various functionals such as volume, surface area, mean width and others. By laws of large numbers, these randomized versions recover the classical inequalities. We give an overview of when such stochastic dominance arises and its applications in convex geometry and probability.