On exposed points of Lipschitz free spaces

TL;DR

This paper characterizes exposed points of the unit ball in Lipschitz free spaces, linking geometric properties of metric segments to the extremal structure of these spaces.

Contribution

It provides a precise criterion for when molecules are exposed points in Lipschitz free spaces, based on the structure of metric segments.

Findings

Exposed points correspond to singleton metric segments.

The characterization relies on a recent intersection property of Lipschitz free spaces.

The result connects geometric and functional-analytic properties of Lipschitz free spaces.

Abstract

In this note we prove that a molecule $d(x,y)^{-1}(\delta(x)-\delta(y))$ is an exposed point of the unit ball of a Lispchitz free space $\mathcal F(M)$ if and only if the metric segment $[x,y]=\{z \in M \; : \; d(x,y)=d(z,x)+d(z,y) \}$ is reduced to $\{x,y\}$. This is based on a recent result due to Aliaga and Perneck\'a which states that the class of Lipschitz free spaces over closed subsets of M is closed under arbitrary intersections when M has finite diameter.