A note on the rate of convergence for a sequence of random polarizations

Abstract

It was shown by Burchard and Fortier that the expected $L^1$ distance between $f^*$ and $n$ random polarizations of an essentially bounded function $f$ with support in a ball of radius $L$ is bounded by $2dm(B_{2L})||f||_{\infty}n^{-1}$. The purpose of this note is to expand on that result. It is shown that the same expected $L^1$ distance is bounded by $c_nn^{-1}$ with $\limsup_{n\rightarrow \infty}c_n \leq 2^{d+1}||\nabla f||_1$ for every $f \in W_{1,1}(B_L) \cap L^{\infty}(B_L)$. Furthermore, the aforementioned expected $L^1$ distance is $O(n^{-1/q})$ for $f \in L^p(B_L)$ with $p>1$ and $\frac{1}{p} + \frac{1}{q} = 1$. An exponential lower bound is provided for the expected measure of the symmetric difference between the random polarizations of measurable sets and their Schwarz symmetrization. Finally, the rate $n^{-1}$ is shown to be, in a sense, best possible for the random polarizations of measurable sets: the expected symmetric difference between the random polarization of a ball and its corresponding Schwarz symmetrization decays at the rate $n^{-1}$.Comment: 13 pages. Improved the presentation, added rate of convergence estimates for unbounded functions, and made major changes to the section on the random polarization of ball