Distances Between Non-symmetric Convex Bodies and the $$MM^* $$ -estimate

Abstract

Let K, D be n-dimensional convex bodes. Define the distance between K and D as $$d(K,D) = \inf \{ \lambda |TK \subset D + x \subset \lambda \cdot TK\} ,$$ where the infimum is taken over all \(x \in {\mathbb{R}}^n \) and all invertible linear operators T. Assume that 0 is an interior point of K and define $$M(K) = \smallint _{S^{n - 1} } |\omega |_K d\mu (\omega ),$$ where μ is the uniform measure on the sphere. We use the difference body estimate to prove that K can be embedded into \({\mathbb{R}}^n \) so that $$M(K) \cdot M(K^ \circ ) \leqslant Cn^{1/3} \log ^a n$$ for some absolute constants C and \(a\). We apply this result to show that the distance between two n-dimensional convex bodies does not exceed \(n^{4/3} \) up to a logarithmic factor.