Strengthened inequalities for the mean width and the $\ell$-norm of origin symmetric convex bodies

TL;DR

This paper establishes near-optimal stability inequalities for the mean width and $\,\ell$-norm of symmetric convex bodies, extending known extremal properties of the cube and crosspolytope.

Contribution

It provides stronger stability versions of classical inequalities involving the mean width and $\,\ell$-norm for symmetric convex bodies, with new results on convex hulls of isotropic measures.

Findings

Proves near-optimal stability for the cube maximizing mean width.

Shows the regular crosspolytope minimizes mean width under certain conditions.

Extends results to the $\,\ell$-norm based on Gaussian integrals.

Abstract

Barthe, Schechtman and Schmuckenschl\"ager proved that the cube maximizes the mean width of symmetric convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball, and the regular crosspolytope minimizes the mean width of symmetric convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. Here we prove close to be optimal stronger stability versions of these results, together with their counterparts about the $\ell$-norm based on Gaussian integrals. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of even isotropic measures on the unit sphere.