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.
Citation
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
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