John’s decomposition in the general case and applications

We give a description of an affine mapping T involving con-tact pairs of two general convex bodies K and L, when T (K) is in a position of maximal volume in L. This extends the classi-cal 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