Supports and extreme points in Lipschitz-free spaces

Abstract

For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}(M)$ are precisely the elementary molecules $(\delta (p)-\delta (q))/d(p,q)$ 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 $r\in M$ 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 $\mathcal{F}(M)$.

arXiv commentv3: Final version. Corrected an embarrassing mistake in the definition of strongly exposed point

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• Extreme points in Lipschitz-free spaces over compact metric spaces
• Convex integrals of molecules in Lipschitz-free spaces
• Projecting Lipschitz functions onto spaces of polynomials
• Delta-points and their implications for the geometry of Banach spaces
• De Leeuw representations of functionals on Lipschitz spaces
• Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces
• Isometries of Lipschitz-free Banach spaces
• Analysis on Laakso graphs with application to the structure of transportation cost spaces