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arXiv · 1206.2041

Convergence in shape of Steiner symmetrizations

Gabriele Bianchi·Almut Burchard·Paolo Gronchi·Aljosa Volcic
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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.

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Gabriele Bianchi, Almut Burchard, Paolo Gronchi, Aljosa Volcic. 2012-06-10. Convergence in shape of Steiner symmetrizations. https://doi.org/10.1512/iumj.2012.61.5087

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