2003/12/02 by Bo’az Klartag, B. Klartag, Klartag, B.
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.FA #math.MG
paper · pdf · doi:10.48550/arxiv.math/0312064
Accepted for publication in Geom. Funct. Anal
arxiv created 2003/12/02 · openalex publication_date 2003/12/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is a classical fact, that given an arbitrary n-dimensional convex body, 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, our technique involves spherical harmonics. We also make use of an earlier result by the author regarding ``isomorphic Minkowski symmetrization''.