On explicit representations of isotropic measures in John and Löwner positions

Abstract

Given a convex body $K\subseteq {\mathbb{R}}^n$ in Löwner 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 ${\frac{\partial }{\partial r}(A_r,v_r)|}_{r=1}$, 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.

19 pages, 1 figure, comments are welcome