Random symmetrizations of measurable sets

TL;DR

This paper proves that sequences of random Steiner symmetrizations of bounded measurable sets almost surely converge to a ball in the Nikodym metric, extending known results from convex bodies to more general sets.

Contribution

It extends convergence results of Steiner symmetrizations from convex bodies to bounded measurable sets in the Nikodym metric.

Findings

Sequences of random Steiner symmetrizations converge almost surely to a ball

Convergence occurs in the Nikodym metric

Results generalize previous work on convex bodies

Abstract

In this paper we prove almost sure convergence to the ball, in the Nikodym metric, of sequences of random Steiner symmetrizations of bounded Caccioppoli and bounded measurable sets, paralleling a result due to Mani-Levitska concerning convex bodies.