# On the preserved extremal structure of Lipschitz-free spaces

## TL;DR

This paper characterizes preserved extreme points in Lipschitz-free spaces using geometric conditions on the underlying metric space, confirming a conjecture relating concavity and metric alignment.

## Contribution

It provides a geometric characterization of preserved extreme points in Lipschitz-free spaces, resolving a conjecture about the relationship between concavity and metric alignment.

## Key findings

- Preserved extreme points correspond to pairs with strictly convex triangle inequalities.
- For compact spaces, the condition simplifies to strict triangle inequality.
- Confirms Weaver's conjecture linking concavity and absence of metrically aligned triples.

## Abstract

We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, each preserved extreme point corresponds to a pair of points $p,q$ in $X$ such that the triangle inequality $d(p,q)\leq d(p,r)+d(q,r)$ is uniformly strict for $r$ away from $p,q$. For compact $X$, this condition reduces to the triangle inequality being strict. This result gives an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points.