Characterizing Lipschitz images of injective metric spaces

TL;DR

This paper characterizes when metric spaces can be represented as Lipschitz images of injective spaces, linking this property to Lipschitz connectedness and intrinsic metric properties.

Contribution

It provides a complete characterization of Lipschitz images of injective metric spaces based on Lipschitz connectedness and metric boundedness.

Findings

Lipschitz images of injective spaces are characterized by Lipschitz connectedness.

Compact Lipschitz images correspond to compact, Lipschitz connected spaces with totally bounded intrinsic metrics.

Separable Lipschitz images are characterized by being analytic, Lipschitz connected, with separable intrinsic metrics.

Abstract

A metric space $X$ is {\em injective} if every non-expanding map $f:B\to X$ defined on a subspace $B$ of a metric space $A$ can be extended to a non-expanding map $\bar f:A\to X$. We prove that a metric space $X$ is a Lipschitz image of an injective metric space if and only if $X$ is Lipschitz connected in the sense that for every points $x,y\in X$, there exists a Lipschitz map $f:[0,1]\to X$ such that $f(0)=x$ and $f(1)=y$. In this case the metric space $X$ carries a well-defined intrinsic metric. A metric space $X$ is a Lipschitz image of a compact injective metric space if and only if $X$ is compact, Lipschitz connected and its intrinsic metric is totally bounded. A metric space $X$ is a Lipschitz image of a separable injective metric space if and only if $X$ is a Lipschitz image of the Urysohn universal metric space if and only if $X$ is analytic, Lipschitz connected and its intrinsic metric is separable.