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 $$ d(K,D) = \inf \{\lambda | T K \subset D+x \subset \lambda \cdot TK \}, $$ where the infimum is taken over all $x \in R^n$ and all invertible linear operators $T$. Assume that 0 is an interior point of $K$ and define $$ M(K) =\int_{S^{n-1}} \| \omega \|_K d \mu (\omega), $$ where $\mu$ is the uniform measure on the sphere. Let $K^{\circ}$ be the polar body of $K$. We use the difference body estimate to prove that $K$ can be embedded into $R^n$ so that $$ M(K) \cdot M(K^{\circ}) \le C n^{1/3} \log^a n $$ 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.