A characterisation of the Daugavet property in spaces of Lipschitz functions

By Luis García-Lirola, Antonín Procházka, Abraham Rueda Zoca. Journal of Mathematical Analysis and Applications, 2017

67 citations recorded by the source corpus.

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 $M$.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0ae68d47420b6769d5d3ae3a325d1735a187eccb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Miguel Martín

In Geometry and Topology

1. B. Cascales
2. Abraham Rueda Zoca
3. Luis García-Lirola
4. Rafael Chiclana
5. Miguel Martín

In Mathematical Physics

1. B. Cascales
2. Abraham Rueda Zoca
3. C. Petitjean
4. Antonín Procházka
5. Luis García-Lirola
6. Rafael Chiclana
7. Miguel Martín

Citations in the corpus

• On strongly norm attaining Lipschitz maps (2018, background): “On the other hand, length spaces have been recently considered in [21] where it is proved that a metric space M is a length space if, and only if, Lip0(M,R) has the Daugavet property [21, Theorem 3.”
• Extremal Structure and Duality of Lipschitz Free Spaces (2017, background): “4 in [12], μ is a strongly exposed point.”