A characterisation of octahedrality in Lipschitz-free spaces

TL;DR

This paper characterizes when Lipschitz-free spaces have an octahedral norm by linking it to a new geometric property of the underlying metric space, revealing surprising embedding limitations.

Contribution

It introduces a new geometric property that characterizes octahedrality in Lipschitz-free spaces and explores its implications for metric space embeddings.

Findings

Metric spaces without the property cannot embed isometrically into  and similar Banach spaces.

The paper establishes a geometric criterion for octahedrality in Lipschitz-free spaces.

Surprising restrictions on embeddings for certain metric spaces.

Abstract

We characterise the octahedrality of Lipschitz-free space norm in terms of a new geometric property of the underlying metric space. We study the metric spaces with and without this property. Quite surprisingly, metric spaces without this property cannot embed isometrically into $\ell_1$ and similar Banach spaces.