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 F(M). We then use this concept to study the extremal structure of F(M). We prove in particular that (δ(x)−δ(y))/d(x,y) is an exposed point of the unit ball of 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 characterize the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of M.