Convergence in shape of Steiner symmetrizations

TL;DR

This paper investigates the convergence behavior of sequences of Steiner symmetrizations of sets in R^n, showing that certain sequences converge in shape even if they do not converge in the traditional sense, and explores various types of direction sequences.

Contribution

It proves convergence in shape for sequences of Steiner symmetrizations with square-summable differences and analyzes convergence for uniformly distributed and finite-direction sequences.

Findings

Sequences with square-summable differences in directions converge in shape.

Convergence in shape does not necessarily lead to an ellipsoid or convex set.

Results extend understanding of Steiner symmetrization behavior for different direction sequences.

Abstract

There are sequences of directions such that, given any compact set K in R^n, the sequence of iterated Steiner symmetrals of K in these directions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of directions which are dense in S^{n-1}.) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.