Extremal cross-polytopes and Gaussian vectors

TL;DR

This paper investigates the extremal properties of cross-polytopes and Gaussian vectors, establishing bounds on mean width and deriving implications for convex bodies and Gaussian distributions.

Contribution

It introduces new bounds on the mean width of cross-polytopes and convex bodies, linking geometric properties to Gaussian vector behavior.

Findings

Minimal mean width occurs when axes are equal in length.

Maximal mean width is achieved when the polytope is two-dimensional.

Provides a lower bound on convex body mean width based on inner radii.

Abstract

Let C = C(l_1, ..., l_n) be the n-dimensional orthogonal cross-polytope whose axes are of length l_1,..., l_n. Subject to the condition \sum l_i^2 = 1, the mean width of C is minimised when l_i = 1/sqrt{n} for every i, and it is maximised when C is at most two dimensional. As a corollary, a lower bound on the mean width of a general convex body K is derived in terms of the successive inner radii of K. A more general result is presented for Gaussian random vectors.