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 $x\ne y\in M$.

Improved main results with respect to v1

References in corpus (3)

• Supports in Lipschitz-free spaces and applications to extremal structure
• On strongly norm attaining Lipschitz maps
• Supports and extreme points in Lipschitz-free spaces

Cited by in corpus (7)

• Supports in Lipschitz-free spaces and applications to extremal structure
• Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
• On relations between transportation cost spaces and ${\ell }_1$
• Extreme points in Lipschitz-free spaces over compact metric spaces
• Convex integrals of molecules in Lipschitz-free spaces
• Hyperbolic Metric Spaces and Stochastic Embeddings
• Projections in Lipschitz-free spaces induced by group actions