Isomorphic Steiner symmetrization
type: Item
instance of: scholarly article
title: Isomorphic Steiner symmetrization
zbMATH Open document ID: 1034.52008
author: B. Klartag
author: Vitali Milman
published in: Inventiones Mathematicae
publication date: 2003-11-04T00:00:00Z
full work available at URL: http://cds.cern.ch/record/622756
review text: Throughout the last 160 years, Steiner symmetrization has become a major tool for proving various geometric inequalities. It is clear that consecutive Steiner symmetrizations make the body closer to a Euclidean ball. Since many years ago mathematicians have been interested in estimating how many symmetrizations are necessary for that convergence. A tremendous improvement was made by \textit{J. Bourgain}, \textit{J. Lindenstrauss}, and \textit{V. Milman} in 1987; they proved using concentration phenomena techniques that this number was of order \(O(n \log n)\). In this paper the authors sharpen this estimate to \(O(n)\). In particular they prove that there exist \(3n\) Steiner symmetrizations that transform any convex set \(K\) of the \(n\)-dimensional Euclidean space into an isomorphic Euclidean ball; i.e. if \(\operatorname{vol}(K)= \operatorname{vol} (D_n)\), where \(D_n\) is the standard Euclidean unit ball, then \(K\) can be transformed into a body \(K'\) such that \(c_1D_n\subset K' \subset c_2D_n\), where \(c_1\) and \(c_2\) are numerical constants. They also prove that for any constant \(c\), arbitrarily close to the value \(2\), \(cn\) symmetrizations are also enough. In addition they prove that for any \(\varepsilon >0\), only \(\lfloor (1+\varepsilon)n \rfloor\) symmetrizations are necessary in order to transform an arbitrary body into an isomorphic ellipsoid. The authors first present a nonoptimal but more accessible proof in two steps: 1) they prove that the number of symmetrizations is proportional to the dimension; 2) they study the symmetrization of two specific bodies: the cube and the cross-polytope. Later they obtain the best constants known up to now using Milman's ``quotient of subspace theorem''.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 2001025
zbMATH Keywords: Steiner symmetrization
zbMATH Keywords: ellipsoid
zbMATH Keywords: \(L_p\)-spaces
MaRDI profile type: Publication
OpenAlex ID: W2066851541
DOI: 10.1007/S00222-003-0290-Y
description: scientific article; zbMATH DE number 2001025