Weakly almost square Lipschitz-free spaces

TL;DR

This paper constructs a novel Lipschitz-free space that is locally almost square but not weakly almost square, explores metric characterizations, and establishes limitations on diameter properties.

Contribution

It provides the first example of a Lipschitz-free space with specific local geometric properties and links metric space structures to Banach space geometry.

Findings

Constructed a Lipschitz-free space that is locally almost square but not weakly almost square.

Proposed a potential metric characterization for weakly almost square Lipschitz-free spaces.

Proved that Lipschitz-free spaces cannot have the symmetric strong diameter 2 property.

Abstract

We construct a Lipschitz-free space that is locally almost square but not weakly almost square; this is the first example of such a Banach space. We also prove a result, which indicates that geodesic metric spaces are a potential metric characterization for weakly almost square Lipschitz-free spaces. Lastly, we prove that a Lipschitz-free space can not have the symmetric strong diameter $2$ property.