On the extremal distance between two convex bodies

We consider the following modified version of the Banach-Mazur distance of convex bodies in \(\mathbb{R}^n :d\left( {K,L} \right) = \inf \left\{ {\left| \lambda \right|:\lambda \in \mathbb{R},\tilde K \subset \tilde L \subset \lambda \tilde K} \right\}\), where the infimum is taken over all non-degenerate affine images \(\tilde K\) and \(\tilde L\) of K and L. Gordon, Litvak, Meyer and Pajor in 2004 showed that for any two convex bodies d(K,L) ≤ n, moreover, if K is a simplex and L = −L then d(K,L) = n. The following question arises naturally: Is equality only attained when one of the sets is a simplex? Leichtweiss in 1959, and later Palmon in 1992 proved that if d(K,B2n) = n, where B2n is the Euclidean ball, then K is the simplex. We prove the affirmative answer to the question in the case when one of the bodies is strictly convex or smooth, thus obtaining a generalization of the result of Leichtweiss and Palmon.