Random symmetrizations of convex bodies

TL;DR

This paper studies the long-term behavior of sequences of Steiner and Minkowski symmetrizations of convex bodies, establishing convergence properties and rates under various conditions.

Contribution

It provides an equivalence result for convergence between Minkowski and Steiner symmetrizations and proves almost sure exponential and subexponential convergence rates.

Findings

Minkowski symmetrizations converge exponentially almost surely.

Steiner symmetrizations converge at rate e^{-c√n} almost surely.

An equivalence between convergence of Minkowski and Steiner symmetrizations is established.

Abstract

In this paper, the asymptotic behavior of sequences of successive Steiner and Minkowski symmetrizations is investigated. We state an equivalence result between the convergences of those sequences for Minkowski and Steiner. Moreover, in the case of independent (and not necessarily identically distributed) directions, we prove the almost sure convergence of successive symmetrizations at rate exponential for Minkowski, and at rate $e^{-c\sqrt{n}}$, with $c>0$, for Steiner.