We characterize preserved extreme points of Lipschitzfree spaces 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) ≤ 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.2010 Mathematics Subject Classification. Primary 46B20; Secondary 46E15, 54E45.

For a complete metric space M , we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space F(M ) are precisely the elementary molecules (δ(p) − δ(q))/d(p, q) defined by pairs of points p, q in M such that the triangle inequality d(p, q) < d(p, r) + d(q, r) is strict for any r ∈ M different from p and q. To this end, we show that the class of Lipschitz-free spaces over closed subsets of M is closed under arbitrary intersections when M has finite diameter, and that this allows a natural definition of the support of elements of F(M ).

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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 characterise the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of M .2010 Mathematics Subject Classification. Primary 46B20; Secondary 46B04, 54E50.