On the Banach-Mazur distance to cross-polytope

TL;DR

This paper demonstrates that the Banach-Mazur distance between the standard cross-polytope and a typical random polytope in high dimensions can grow faster than previously known, indicating the cross-polytope is not an approximate center of the Minkowski compactum.

Contribution

It establishes a new lower bound on the Banach-Mazur distance to the cross-polytope using random polytopes, improving prior estimates and showing the cross-polytope's limitations as an approximate center.

Findings

Banach-Mazur distance grows at least as n^{5/9} log^{-C} n

Random polytopes with 2n^C vertices are far from the cross-polytope

Cross-polytope cannot serve as an approximate center of the Minkowski compactum

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.