  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.
