Embeddings of Lipschitz-free spaces into $\ell_1$

TL;DR

This paper characterizes when Lipschitz-free spaces over metric spaces embed into , linking geometric properties of the space with the embedding, and describes the extreme points of their unit balls.

Contribution

It provides necessary and sufficient conditions for isometric and almost-isometric embeddings of Lipschitz-free spaces into , based on the structure of the underlying metric space as an -tree.

Findings

Embedding into  occurs iff the space is a subset of an -tree with negligible length measure.

Isometric embedding requires the length measure of the closure of branching points to be negligible.

Extreme points of the unit ball are characterized explicitly as differences of Dirac measures normalized by distance.

Abstract

We show that, for a separable and complete metric space $M$, the Lipschitz-free space $\mathcal F(M)$ embeds linearly and almost-isometrically into $\ell_1$ if and only if $M$ is a subset of an $\mathbb R$-tree with length measure 0. Moreover, it embeds isometrically if and only if the length measure of the closure of the set of branching points of $M$ (taken in any minimal $\mathbb R$-tree that contains $M$) is negligible. We also prove that, for any subset $M$ of an $\mathbb R$-tree, every extreme point of the unit ball of $\mathcal F(M)$ is an element of the form $(\delta(x)-\delta(y))/d(x,y)$ for $x\neq y\in M$.