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 $M$.

arXiv commentFinal accepted version. Only cosmetic changes wrt v1

References in corpus (6)

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• Generalized transportation cost spaces
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Cited by in corpus (16)

• Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
• Embeddings of Lipschitz-free spaces into ${\ell }_1$
• Integral representation and supports of functionals on Lipschitz spaces
• Points of differentiability of the norm in Lipschitz-free spaces
• Extreme points in Lipschitz-free spaces over compact metric spaces
• Convex integrals of molecules in Lipschitz-free spaces
• Delta-points and their implications for the geometry of Banach spaces
• De Leeuw representations of functionals on Lipschitz spaces
• Normal functionals on Lipschitz spaces are weak${}^{\ast }$ continuous
• Pełczyński's property (V${}^{\ast }$) in Lipschitz-free spaces
• Tilings of the Hyperbolic Space and Lipschitz Functions
• Isometries of Lipschitz-free Banach spaces
• Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces
• Injectivity of Lipschitz operators
• On the weak${}^{\ast }$ separability of the space of Lipschitz functions
• A note on the spectrum of Lipschitz operators and composition operators on Lipschitz spaces