On explicit representations of isotropic measures in John and Löwner positions

TL;DR

The paper addresses the explicit construction of a nonnegative centered isotropic measure supported on the contact points of a convex body K in Löwner position, a key ingredient in John/Löwner theory. It introduces a finite-dimensional convex minimization via the functionals I_ν and L_r, showing that the minimizer yields an isotropic measure through F′(⟨ξ, M_0ξ + w_0⟩) dν(ξ). By connecting to Artstein-Avidan and Katzin’s maximal intersection positions, it also analyzes the r→1 limit, defining I_1 and proving convergence of minimizers (A_r,v_r) to (Id,0) and the corresponding derivative to a tangent isotropic data. The results establish equivalence criteria linking the existence of a minimizer to the interior of the convex hull of contact data, providing a constructive route to isotropic measures in John/Löwner geometry with potential applications to reverse isoperimetric inequalities and related convex-geometric problems.

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 $\left. \frac{\partial}{\partial r}(A_r, v_r)\right|_{r=1}$, 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.

Figures (2)

• Figure 1: functions $f$ (blue) and $g$ (orange),
• Figure 2: functions $f_r$ (blue) and $g_r$ (orange)

Theorems & Definitions (25)

• Theorem 1
• Theorem 2: Theorem 1.6, artstein2018isotropic
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Definition 10