On the Banach-Mazur distance to cross-polytope

Let n >= 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 G(m) in R-n such that the Banach-Mazur distance dBm (B-1(n), G(m)) satisfies d(BM)(B-1(n), G(m)) >= n(5/9) log(-C) n, where C > 0 is a universal constant. The body G(m) is obtained as a typical realization of a random polytope in R-n with 2m := 2n(3) vertices. The result improves upon an earlier estimate of S. Szarek which gives d(BM)(B-1(n), G(m)) >= cn(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. (C) 2019 Elsevier Inc. All rights reserved.