Supports in Lipschitz-free spaces and applications to extremal structure

TL;DR

This paper advances the understanding of Lipschitz-free spaces by establishing closure under intersections, defining supports, and characterizing extremal points, with implications for the geometric structure of these spaces.

Contribution

It introduces a general support concept for Lipschitz-free spaces and characterizes their extremal structure, extending previous finite-diameter results.

Findings

Lipschitz-free spaces are closed under arbitrary intersections.

Exposed points are characterized by trivial metric segments.

Extreme points of the positive unit ball are evaluation functionals.

Abstract

We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space $M$ is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space $\mathcal F(M)$. We then use this concept to study the extremal structure of $\mathcal F(M)$. We prove in particular that $(\delta(x) - \delta(y))/d(x,y)$ is an exposed point of the unit ball of $\mathcal F(M)$ whenever the metric segment $[x,y]$ is trivial, and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself. We also characterise the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of $M$.