Isotropic Measures and Maximizing Ellipsoids: Between John and Loewner

TL;DR

This paper introduces a family of convex body positions interpolating between John and Loewner positions, establishing a connection with isotropic measures and providing new insights into classical geometric theorems.

Contribution

It defines a parameterized family of positions for convex bodies that interpolate between John and Loewner positions, linking these to isotropic measures and offering a new perspective on contact point theorems.

Findings

Each position induces an isotropic measure on the sphere.

Maximal intersection position of a specific radius yields an M-position with an associated isotropic measure.

John's theorem contact points can be viewed as a limit case of these measures.

Abstract

We define a one parameter family of positions of a convex body which interpolates between the John position and the Loewner position: for $r>0$, we say that $K$ is in maximal intersection position of radius $r$ if $\textrm{Vol}_{n}(K\cap rB_{2}^{n})\geq \textrm{Vol}_{n}(K\cap rTB_{2}^{n})$ for all $T\in SL_{n}$. We show that under mild conditions on $K$, each such position induces a corresponding isotropic measure on the sphere, which is simply a normalized Lebesgue measure on $r^{-1}K\cap S^{n-1}$. In particular, for $r_{M}$ satisfying $r_{M}^{n}\kappa_{n}=\textrm{Vol}_{n}(K)$, the maximal intersection position of radius $r_{M}$ is an $M$-position, so we get an $M$-position with an associated isotropic measure. Lastly, we give an interpretation of John's theorem on contact points as a limit case of the measures induced from the maximal intersection positions.