Embeddings of Lipschitz-free spaces into $\ell_1$

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 $x\neq y\in M$.