Characterizing Lipschitz images of injective metric spaces

A metric space X is { injective} if every non-expanding map f:B→ X defined on a subspace B of a metric space A can be extended to a non-expanding map f:A→ 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∈ X, there exists a Lipschitz map f:[0,1]→ 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.