Convergence in shape of Steiner symmetrizations

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.

11 pages

Cited by in corpus (6)

• Convergence of symmetrization processes
• Convergence properties of symmetrization processes
• Approximation of symmetrizations by Markov processes
• Steiner symmetrization along a certain equidistributed sequence of directions
• The Petty projection inequality for sets of finite perimeter
• Generalization of Klain's Theorem to Minkowski Symmetrization of compact sets and related topics