Extreme points in Lipschitz-free spaces over compact metric spaces

TL;DR

This paper characterizes the extreme points of the unit ball in Lipschitz-free spaces over compact metric spaces, showing they all have finite support and are also extreme in the bidual ball, advancing understanding of their geometric structure.

Contribution

It provides a complete characterization of extreme points in Lipschitz-free spaces over compact metric spaces, linking finite support with extremality in the bidual.

Findings

All extreme points have finite support.

Extreme points are also extreme in the bidual ball.

Develops properties of an integral representation of functionals.

Abstract

We prove that all extreme points of the unit ball of a Lipschitz-free space over a compact metric space have finite support. Combined with previous results, this completely characterizes extreme points and implies that all of them are also extreme points in the bidual ball. For the proof, we develop some properties of an integral representation of functionals on Lipschitz spaces originally due to K. de Leeuw.