Daugavet points and $\Delta $-points in Lipschitz-free spaces
Volume 265 / 2022
Studia Mathematica 265 (2022), 37-55
MSC: Primary 46B04; Secondary 46B20.
DOI: 10.4064/sm210111-5-5
Published online: 14 February 2022
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 $2$ from $\mu $. Moreover, we prove that if $x$ and $y$ are connectable by rectifiable curves of length 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, 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.
