Daugavet points and $\Delta$-points in Lipschitz-free spaces

TL;DR

This paper investigates the geometric properties of Lipschitz-free Banach spaces, characterizing Daugavet and Δ-points, and providing conditions under which these points exist or coincide, revealing new insights into their structure.

Contribution

The paper provides a complete characterization of Daugavet points in Lipschitz-free spaces over compact metric spaces and explores conditions for Δ-points, highlighting differences between these types of points.

Findings

Daugavet points correspond to the absence of nearby denting points.

Molecules are Δ-points if connectable by nearly shortest rectifiable curves.

Lipschitz-free spaces can have Δ-points that are not Daugavet points.

Abstract

We study Daugavet points and $\Delta$-points in Lipschitz-free Banach spaces. We prove that, if $M$ is a compact metric space, then $\mu\in S_{\mathcal F(M)}$ is a Daugavet point if, and only if, there is no denting point of $B_{\mathcal F(M)}$ at distance strictly smaller than two from $\mu$. Moreover, we prove that if $x$ and $y$ are connectable by rectifiable curves of lenght as close to $d(x,y)$ as we wish, then the molecule $m_{x,y}$ is a $\Delta$-point. Some conditions on $M$ which guarantee that the previous implication reverses are also obtained. As a consequence of our work, we show that Lipschitz-free spaces are natural examples of Banach spaces where we can guarantee the existence of $\Delta$-points which are not Daugavet points.