Supports and extreme points in Lipschitz-free spaces

TL;DR

This paper characterizes the extreme points of the unit ball in Lipschitz-free spaces over complete metric spaces, linking them to elementary molecules defined by pairs of points with strict triangle inequality conditions.

Contribution

It provides a precise description of finitely supported extreme points in Lipschitz-free spaces and introduces a method to define supports of elements in these spaces.

Findings

Finitely supported extreme points are elementary molecules with strict triangle inequality.

Lipschitz-free spaces over closed subsets are closed under intersections for finite diameter spaces.

A natural definition of support for elements in Lipschitz-free spaces is established.

Abstract

For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}(M)$ are precisely the elementary molecules $(\delta(p)-\delta(q))/d(p,q)$ defined by pairs of points $p,q$ in $M$ such that the triangle inequality $d(p,q)<d(p,r)+d(q,r)$ is strict for any $r\in M$ different from $p$ and $q$. To this end, we show that the class of Lipschitz-free spaces over closed subsets of $M$ is closed under arbitrary intersections when $M$ has finite diameter, and that this allows a natural definition of the support of elements of $\mathcal{F}(M)$.