On the Banach–Mazur distance to cross-polytope

Abstract

Let n≥3, and let B1n 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 Gm in Rn such that the Banach–Mazur distance dBM(B1n,Gm) satisfies dBM(B1n,Gm)≥n5/9log−C⁡n, where C>0 is a universal constant. The body Gm is obtained as a typical realization of a random polytope in Rn with 2m:=2n3 vertices. The result improves upon an earlier estimate of S. Szarek which gives dBM(B1n,Gm)≥cn1/2log⁡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.