# A characterisation of the Daugavet property in spaces of Lipschitz functions

## TL;DR

This paper characterizes when spaces of Lipschitz functions and their free spaces have the Daugavet property, linking it to the geometric structure of the underlying metric space, especially length and convexity properties.

## Contribution

It establishes a precise characterization of the Daugavet property in Lipschitz and free Lipschitz spaces based on metric space properties, including length and convexity.

## Key findings

- Lipschitz space has the Daugavet property iff the metric space is a length space.
- For compact spaces, the free Lipschitz space either has the Daugavet property or a strongly exposed point.
- In infinite compact subsets of strictly convex spaces, the Daugavet property is equivalent to the convexity of the set.

## 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 $M$.