Geometry and volume product of finite dimensional Lipschitz-free spaces

TL;DR

This paper explores the geometric properties of Lipschitz-free spaces over finite metric spaces, characterizing extremal cases for volume products and identifying conditions for maximal and minimal volume configurations.

Contribution

It provides new characterizations of when Lipschitz-free spaces form Hanner polytopes and identifies conditions for extremal volume products in finite metric spaces.

Findings

Characterization of metric spaces with Hanner polytope unit balls

Identification of conditions for maximal volume product

Analysis of metric spaces minimizing volume product

Abstract

The goal of this paper is to study geometric and extremal properties of the convex body $B_{\mathcal F(M)}$, which is the unit ball of the Lipschitz-free Banach space associated with a finite metric space $M$. We investigate $\ell_1$ and $\ell_\infty$-sums, in particular we characterize the metric spaces such that $B_{\mathcal F(M)}$ is a Hanner polytope. We also characterize the finite metric spaces whose Lipschitz-free spaces are isometric. We discuss the extreme properties of the volume product $\mathcal{P}(M)=|B_{\mathcal F(M)}|\cdot|B_{\mathcal F(M)}^\circ|$, when the number of elements of $M$ is fixed. We show that if $\mathcal P(M)$ is maximal among all the metric spaces with the same number of points, then all triangle inequalities in $M$ are strict and $B_{\mathcal F(M)}$ is simplicial. We also focus on the metric spaces minimizing $\mathcal P(M)$, and in the Mahler's conjecture for this class of convex bodies.