# 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 language English 829-843 15 Combinatorics Probability and Computing 14 5-6 https://doi.org/10.1017/S0963548305006899 Published - Nov 2005

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