Diameter two properties for spaces of Lipschitz functions

TL;DR

This paper investigates various diameter two properties in Banach spaces of Lipschitz functions, clarifying their distinctions and establishing new results about these properties and Daugavet points using the de Leeuw transform.

Contribution

It distinguishes between different diameter two properties in Lipschitz function spaces and proves new results about their presence and characteristics, including the case of $ ext{Lip}_0(K_n)$ spaces.

Findings

The diameter two, strong diameter two, and symmetric strong diameter two properties are all different in these spaces.

$ ext{Lip}_0(K_n)$ has the symmetric strong diameter two property for all $n$.

Every local norm-one Lipschitz function is a Daugavet point.

Abstract

We solve some open problems regarding diameter two properties within the class of Banach spaces of real-valued Lipschitz functions by using the de Leeuw transform. Namely, we show that: the diameter two property, the strong diameter two property, and the symmetric strong diameter two property are all different for these spaces of Lipschitz functions; the space $\operatorname{Lip}_0(K_n)$ has the symmetric strong diameter two property for every $n\in \mathbb{N}$, including the case of $n=2$; every local norm-one Lipschitz function is a Daugavet point.