John’s Decomposition in the General Case and Applications

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 Rn have non-degenerate affine images K ′ and L′ such that K ′ ⊂ L′ ⊂ −nK ′. 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′′ ⊂ nK ′′. 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. 2000 Mathematical Subject Classification. Primary: 46B, 52A. Secondary: 51M16, 53A55.