Isomorphic Steiner symmetrization
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''.
scientific article; zbMATH DE number 2001025