On the Banach-Mazur distance to cross-polytope

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 ${\mathbb R}^n$suchthattheBanach--Mazurdistance$d(B_1^n,\mathcal G_m)$satisfies$d(B_1^n,\mathcal G_m)\geq n^{5/9}\log^{-C}n$,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 $d(B_1^n,{\mathcal{G}}_m)\ge c{n}^{1\mathrm{/}2}\log n$ (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.