Rapid Steiner symmetrization of most of a convex body and the slicing problem

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Abstract

For an arbitrary n-dimensional convex body, at least almost n Steiner symmetrizations are required in order to symmetrize the body into an isomorphic ellipsoid. We say that a body T ⊂ ℝn is 'quickly symmetrizable with function $c(\varepsilon)$' if for any $\varepsilon > 0$ there exist only $\lfloor \varepsilon n \rfloor$ symmetrizations that transform T into a body which is $c(\varepsilon)$-isomorphic to an ellipsoid. In this note we ask, given a body K ⊂ ℝn, whether it is possible to remove a small portion of its volume and obtain a body $T \subset K$ which is quickly symmetrizable. We show that this question, for $c(\varepsilon) $ polynomially depending on 1\ε, is equivalent to the slicing problem.

Original languageEnglish
Pages (from-to)829-843
Number of pages15
JournalCombinatorics Probability and Computing
Volume14
Issue number5-6
DOIs
StatePublished - Nov 2005

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