The Lipschitz-free space over length space is locally almost square but never almost square

TL;DR

This paper investigates the geometric properties of Lipschitz-free spaces over length metric spaces, establishing conditions under which they are locally almost square and relating this to the Daugavet property, while also proving they are never almost square.

Contribution

It proves that Lipschitz-free spaces over length spaces are locally almost square and characterizes when they have the Daugavet property, also showing they are never almost square.

Findings

Lipschitz-free spaces over length spaces are locally almost square.

Such spaces have the Daugavet property if and only if they are locally almost square.

Lipschitz-free spaces are never almost square.

Abstract

We prove that the Lipschitz-free space over a metric space M is locally almost square whenever M is a length space. Consequently, the Lipschitz-free space is locally almost square if and only if it has the Daugavet property. We also show that a Lipschitz-free space is never almost square.