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

Abstract

We study Daugavet points and ∆-points in Lipschitz-free Banach spaces. We prove that if M is a compact metric space, then µ ∈ S F (M ) is a Daugavet point if and only if there is no denting point of B F (M ) at distance strictly smaller than 2 from µ. 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 mx,y is a ∆-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 ∆-points which are not Daugavet points.

Asymptotic geometry and delta-points

“…x − θe t JT ≤ x − x t e t JT + (1 − α) ≤ 2 − α so that x is at distance strictly less than 2 from a (weak * ) denting point of JT . The conclusion follows from a weak * version of [30,Proposition 3.1]. Now we will show that JT does not admit ∆-points.…”

“…Can molecules in JT * be ∆-points? Note that the observation from [30,Remark 2.4] applied to the set of molecules in JT * which satisfies as mentioned conv M = B X * gives the following simplification. Although it is possible to do some computations in very specific cases (for example when x * is a molecule supported on two segments) it seems to be difficult to estimate the distance between two molecules in general even when they are supported on the same branch of T .…”

“…Another striking example illustrating this was obtained in the context of Lipschitz-free spaces. The study of Daugavet-and ∆-points in this context started in [30] where a characterization of Daugavet-points in free-spaces over compact metric spaces was discovered. It was observed that Daugavetpoints have to be at distance 2 from every denting point of the unit ball in any given Banach space, [30,Proposition 3.1], and it was proved that the converse holds in every free-space in the compact setting.…”

“…The study of Daugavet-and ∆-points in this context started in [30] where a characterization of Daugavet-points in free-spaces over compact metric spaces was discovered. It was observed that Daugavetpoints have to be at distance 2 from every denting point of the unit ball in any given Banach space, [30,Proposition 3.1], and it was proved that the converse holds in every free-space in the compact setting. Extending this result to the general setting, Veeorg was then able to provide in [46] a surprising example of a metric space whose free-space has the Radon-Nikodým property (RNP) and admits a Daugavet-point.…”

Daugavet points in projective tensor products

“…Therefore, the concept of Daugavet points can be faced as a local version of the Daugavet property. See [1,2,7,9] for background on Daugavet points and diverse relevant examples.…”