Random polarizations

TL;DR

This paper establishes conditions for the almost sure convergence of random polarization sequences to symmetric decreasing rearrangements, providing bounds on convergence rates and exploring applications to Steiner symmetrizations.

Contribution

It introduces new convergence criteria for random polarizations with non-uniform distributions and applies these to Steiner symmetrizations without convexity assumptions.

Findings

Convergence of random polarizations to symmetric rearrangements under broad conditions.

Bounds on the rate of convergence for Steiner symmetrizations.

Examples where convergence fails or is slow.

Abstract

We derive conditions under which random sequences of polarizations (two-point symmetrizations) converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose distributions need not be uniform. The proof of convergence hinges on an estimate for the expected distance from the limit that also yields a bound on the rate of convergence. In the special case of i.i.d. sequences, we obtain almost sure convergence even for polarizations chosen at random from suitable small sets. As corollaries, we find bounds on the rate of convergence of Steiner symmetrizations that require no convexity assumptions, and show that full rotational symmetry can be achieved by randomly alternating Steiner symmetrization in a finite number of directions that satisfy an explicit non-degeneracy condition. We also present some negative results on the rate of convergence and give examples where convergence fails.