Rate of convergence of random polarizations

TL;DR

This paper investigates the rate at which random polarizations of a set on a sphere converge to a symmetric shape, establishing the optimality of the convergence rate and how the constant depends on the dimension.

Contribution

It proves that the convergence rate of 1/n is optimal and that the constant involved grows at least linearly with the dimension.

Findings

The expected symmetric difference decreases at a rate of 1/n.

The constant in the bound grows at least linearly with dimension.

The power law convergence rate is proven to be optimal.

Abstract

After n random polarizations of Borel set on a sphere, its expected symmetric difference from a polar cap is bounded by C/n, where the constant depends on the dimension [arXiv:1104.4103]. We show here that this power law is best possible, and that the constant grows at least linearly with the dimension.