Supports in Lipschitz-free spaces and applications to extremal structure

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 $M$.

Final accepted version. Only cosmetic changes wrt v1