Randomized Isoperimetric Inequalities

Abstract

We discuss isoperimetric inequalities for convex sets. These include the clas-sical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santaló, Busemann-Petty and their various extensions. We show that many such in-equalities 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 inequal-ities. We give an overview of when such stochastic dominance arises and its applications in convex geometry and probability.