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.