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arXiv · 1705.05145

A characterisation of the Daugavet property in spaces of Lipschitz functions

Luis García-Lirola·Antonín Procházka·Abraham Rueda Zoca
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Abstract

We study the Daugavet property in the space of Lipschitz functions $\operatorname{Lip}_0(M)$ for a complete metric space $M$. Namely we show that $\operatorname{Lip}_0(M)$ has the Daugavet property if and only if $M$ is a length space. This condition also characterises the Daugavet property in the Lipschitz free space $\mathcal{F}(M)$. Moreover, when $M$ is compact, we show that either $\mathcal{F}(M)$ has the Daugavet property or its unit ball has a strongly exposed point. If $M$ is an infinite compact subset of a strictly convex Banach space then the Daugavet property of $\operatorname{Lip}_0(M)$ is equivalent to the convexity of $M$.

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Luis García-Lirola, Antonín Procházka, Abraham Rueda Zoca. 2017-05-15. A characterisation of the Daugavet property in spaces of Lipschitz functions. https://arxiv.org/abs/1705.05145

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