On the Banach-Mazur distance to cross-polytope

Abstract: Let $n\geq 3$, 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 $d(B_1^n,\mathcal G_m)\geq n^{5/9}\log^{-C}n$, 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 $d(B_1^n,\mathcal G_m)\geq c n^{1/2}\log n$ (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.