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arXiv · 1705.09579

On the preserved extremal structure of Lipschitz-free spaces

Ramón J. Aliaga·Antonio J. Guirao
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Abstract

We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ 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 $d(p,q)\leq d(p,r)+d(q,r)$ 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.

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Ramón J. Aliaga, Antonio J. Guirao. 2017-05-26. On the preserved extremal structure of Lipschitz-free spaces. https://doi.org/10.4064/sm170529-30-11

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