IUMJ

Title: Convergence in shape of Steiner symmetrizations

Authors: Gabriele Bianchi, Almut Burchard, Paolo Gronchi and Aljosa Volcic

Issue: Volume 61 (2012), Issue 4, 1695-1710

Abstract:

It is known that the iterated Steiner symmetrals of any given compact sets converge to a ball for most sequences of directions. However, examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. Here we show that such sequences converge \emph{in shape}. The limit need not be an ellipsoid or even a convex set. We also consider uniformly distributed sequences of directions, and we extend a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.