Distances between non--symmetric convex bodies and the $M{M}^{\ast }$-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 $x\in R^n$ and all invertible linear operators $T$. Assume that 0 is an interior point of $K$ and define $$M(K)={\int }_{S^{n-1}}\Vert \omega {\Vert }_{K}d\mu (\omega ),$$ 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.

15 pages, AMSTeX