Characterizations of Daugavet- and delta-points in Lipschitz-free spaces

TL;DR

This paper characterizes Daugavet- and delta-points in Lipschitz-free spaces and constructs the first example of such a space with the Radon–Nikodym property containing a Daugavet-point.

Contribution

It provides the first characterization of Daugavet- and delta-points in Lipschitz-free spaces and constructs a novel example of a Lipschitz-free space with the Radon–Nikodym property that contains a Daugavet-point.

Findings

Characterization of Daugavet- and delta-points in Lipschitz-free spaces.

Construction of a Lipschitz-free space with the Radon–Nikodym property containing a Daugavet-point.

First known example of such a space with these properties.

Abstract

A norm one element $x$ of a Banach space is a Daugavet-point (respectively, a $\Delta$-point) if every slice of the unit ball (respectively, every slice of the unit ball containing $x$) contains an element, which is almost at distance 2 from $x$. We characterize Daugavet- and $\Delta$-points in Lipschitz-free spaces. Furthermore, we construct a Lipschitz-free space with the Radon--Nikod\'ym property and a Daugavet-point; this is the first known example of such a Banach space.