On explicit representations of isotropic measures in John and L\"owner positions

TL;DR

This paper investigates explicit constructions of isotropic measures supported on contact points of convex bodies in L"owner position, linking the problem to convex optimization and geometric interpretations related to maximal intersection positions.

Contribution

It introduces a convex optimization approach to construct isotropic measures in L"owner position and provides a geometric interpretation of the minimizer in terms of a family of convex body positions.

Findings

Convex function minimization characterizes isotropic measures.

Geometric interpretation relates to maximal intersection positions.

Method applies to convex bodies in L"owner position.

Abstract

Given a convex body $K \subseteq \mathbb R^n$ in L\"owner position we study the problem of constructing a non-negative centered isotropic measure supported in the contact points, whose existence is guaranteed by John's Theorem. The method we propose requires the minimization of a convex function defined in an $\frac {n(n+3)}2$ dimensional vector space. We find a geometric interpretation of the minimizer as $\left. \frac{\partial}{\partial r}(A_r, v_r)\right|_{r=1}$, where $A_r K + v_r$ is a one-parameter family of positions of $K$ that are in some sense related to the maximal intersection position of radius $r$ defined recently by Artstein-Avidan and Katzin.