TY - JOUR

T1 - Rapid Steiner symmetrization of most of a convex body and the slicing problem

AU - Klartag, B.

AU - Milman, V.

PY - 2005/11

Y1 - 2005/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=26644450289&partnerID=8YFLogxK

U2 - 10.1017/S0963548305006899

DO - 10.1017/S0963548305006899

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AN - SCOPUS:26644450289

SN - 0963-5483

VL - 14

SP - 829

EP - 843

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 5-6

ER -