John's Decomposition in the General Case and Applications

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.