Steiner symmetrization along a certain equidistributed sequence of directions

TL;DR

This paper demonstrates that iterated Steiner symmetrizations along a Van der Corput sequence of directions in the plane converge to the symmetric decreasing rearrangement, using self-similarity and competing symmetries.

Contribution

It introduces a novel approach by using equidistributed Van der Corput sequences for Steiner symmetrizations and proves convergence in this setting.

Findings

Convergence of Steiner symmetrizations along Van der Corput sequence

Utilization of self-similarity and competing symmetries in the proof

Extension of symmetrization techniques to equidistributed directions

Abstract

This note reports the results of an undergraduate research project from the year 2013-14, concerning the convergence of iterated Steiner symmetrizations in the plane. The directions of symmetrization are chosen according to the Van der Corput sequence, a classical example of a sequence that is equidistributed on the unit circle with low discrepancy. It is shown here that the resulting iteration of Steiner symmetrizations converges to the symmetric decreasing rearrangement. The proof exploits the self-similarity of the sequence of angular increments, using the technique of competing symmetries.