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September 2004 John's Decomposition in the General Case and Applications
Y. Gordon, A.E. Litvak, M. Meyer, A. Pajor
J. Differential Geom. 68(1): 99-119 (September 2004). DOI: 10.4310/jdg/1102536711

Abstract

We give a description of an affine mapping T involving contact pairs of two general convex bodies K and L, when T(K) is in a position of maximal volume in L. This extends the classical John's theorem of 1948, and is applied to the solution of a problem of Grünbaum; namely, any two convex bodies K and L in ℝ n have non-degenerate affine images K′ and L′ such that K′ ⊂ L′ ⊂ - n K′. As a corollary, we obtain that if L has a center of symmetry, then there are non-degenerate affine images K″ and L″ of K and L such that K″ ⊂ L″ ⊂ n K″. Other applications to volume ratios and distance estimates are given. In particular, the Banach-Mazur distance between the n-dimensional simplex and any centrally symmetric convex body is equal to n.

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Y. Gordon. A.E. Litvak. M. Meyer. A. Pajor. "John's Decomposition in the General Case and Applications." J. Differential Geom. 68 (1) 99 - 119, September 2004. https://doi.org/10.4310/jdg/1102536711

Information

Published: September 2004
First available in Project Euclid: 8 December 2004

zbMATH: 1120.52004
MathSciNet: MR2152910
Digital Object Identifier: 10.4310/jdg/1102536711

Rights: Copyright © 2004 Lehigh University

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Vol.68 • No. 1 • September 2004
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