On Certain Extremal Banach-Mazur Distances and Ader's Characterization of Distance Ellipsoids

TL;DR

This paper characterizes the extremal cases of Banach-Mazur distances for convex bodies, especially in three dimensions, using Ader's older characterization of distance ellipsoids, and explores related geometric extremal problems.

Contribution

It extends Ader's characterization to prove the uniqueness of maximizers for the Banach-Mazur distance in three dimensions and investigates extremal planar convex bodies.

Findings

The parallelotope and cross-polytope are the unique maximizers in 3D.

Ader's characterization provides a volume-free proof of the $\

Abstract

A classical consequence of the John Ellipsoid Theorem is the upper bound $\sqrt{n}$ on the Banach-Mazur distance between the Euclidean ball and any symmetric convex body in $\mathbb{R}^n$. Equality is attained for the parallelotope and the cross-polytope. While it is known that they are unique with this property for $n=2$ but not for $n \geq 4$, no proof of the characterization of the three-dimensional equality case seems to have ever been published. We fill this gap by showing that the parallelotope and the cross-polytope are the unique maximizers for $n=3$. Our proof is based on an extension of a characterization of distance ellipsoids due to Ader from $1938$, which predates the John Ellipsoid Theorem. Ader's characterization turns out to provide a decomposition similar to the John decomposition, which leads to a proof of the aforementioned $\sqrt{n}$ estimate that bypasses the concept of volumes and reveals precise information about the equality case. We highlight further consequences of Ader's characterization, including a proof of an unpublished result attributed to Maurey related to the uniqueness of distance ellipsoids. Additionally, we investigate more closely the role of the parallelogram as a maximizer in various problems related to the distance between planar symmetric convex bodies. We establish the stability of the parallelogram as the unique planar symmetric convex body with the maximal distance to the Euclidean disc with the best possible (linear) order. This uniqueness extends to the setting of pairs of planar $1$-symmetric convex bodies, where we show that the maximal possible distance between them is again $\sqrt{2}$, together with a characterization of the equality case involving the parallelogram.