Rate of convergence of geometric symmetrizations

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Abstract

It is a classical fact, that given an arbitrary convex body K ⊃ ℝn there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we provide quantitative estimates regarding this convergence, for both Minkowski and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean ball, improving previously known exponential estimates. Inspired by a method of Diaconis [D], our technique involves spherical harmonics. We also make use of an earlier result by the author regarding "isomorphic Minkowski symmetrization".

Original languageEnglish
Pages (from-to)1322-1338
Number of pages17
JournalGeometric and Functional Analysis
Volume14
Issue number6
DOIs
StatePublished - Dec 2004

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