The local diameter two property and the diameter two property in spaces of Lipschitz functions

TL;DR

This paper distinguishes various diameter two properties in Lipschitz function spaces, providing characterizations and introducing a generalized cyclical monotonicity to analyze these properties.

Contribution

It separates the local diameter two property from the diameter two property and their weak-star versions, offering new characterizations and a generalized monotonicity concept.

Findings

Separated local and global diameter two properties in Lipschitz spaces

Characterized $w^*$-LD$2$P, $w^*$-D$2$P, LD$2$P, and SD$2$P

Introduced a generalized cyclical monotonicity for functional analysis

Abstract

We separate the local diameter two property from the diameter two property and their weak-star counterparts from each other in spaces of Lipschitz functions. We characterise the $w^*$-LD$2$P, the $w^*$-D$2$P, the LD$2$P, and the SD$2$P in these spaces. We introduce a generalised version of cyclical monotonicity to study functionals on spaces of Lipschitz functions.