On the Banach-Mazur distance to cross-polytope

Abstract

Let $n\ge 3$, and let $B_1^n$ be the standard $n$-dimensional cross-polytope (i.e. the convex hull of standard coordinate vectors and their negatives). We show that there exists a symmetric convex body ${\mathcal{G}}_m$ in ${\mathbb{R}}^n$ such that the Banach--Mazur distance $d(B_1^n,{\mathcal{G}}_m)$ satisfies $d(B_1^n,{\mathcal{G}}_m)\ge n^{5/9}{\log }^{-C}n$, where $C>0$ is a universal constant. The body ${\mathcal{G}}_m$ is obtained as a typical realization of a random polytope in ${\mathbb{R}}^n$ with $2m:=2n^C$ vertices (for a large constant $C$). The result improves upon an earlier estimate of S.Szarek which gives $d(B_1^n,{\mathcal{G}}_m)\ge c{n}^{1/2}\log n$ (with a different choice of $m$). This shows in a strong sense that the cross-polytope (or the cube $[-1,1{]}^n$) cannot be an "approximate" center of the Minkowski compactum.

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