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

Károly J. Böröczky, Ferenc Fodor, Daniel Hug

Abstract

Barthe, Schechtman and Schmuckenschläger 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 Löwner 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.

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

Abstract

Barthe, Schechtman and Schmuckenschläger 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 Löwner 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 -norm based on Gaussian integrals. We also consider related stability results for the mean width and the -norm of the convex hull of the support of even isotropic measures on the unit sphere.

Paper Structure

This paper contains 7 sections, 31 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. The $\ell$-norm when the Löwner ellipsoid is a ball
  3. Extremal $\ell$-norm when the John ellipsoid is a ball and the rank one Brascamp--Lieb inequality
  4. Auxiliary statements to strengthen Theorem \ref{['ell-sym-John']}
  5. Stability around the cube when the John ellipsoid is a ball
  6. Proof of Theorem \ref{['Lowner-stab']} and Theorem \ref{['John-stab']}
  7. Proof of Theorem \ref{['meanw-iso-measure-stab']}

Key Result

Theorem 1.1

Let $K$ be an origin symmetric convex body in $\mathbb{R}^n$.

Theorems & Definitions (53)

  • Theorem 1.1: Barthe '98, Schechtman, Schmuckenschläger '95
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Li, Leng LiL12
  • Theorem 1.5
  • Theorem 2.1: Barthe, Schechtman, Schmuckenschläger
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 43 more