A uniform-distance-preserving Kirszbraun extension exists iff the reference map satisfies a barycentric inequality, for arbitrary real Hilbert targets and subsets.
Ciosmak,Continuity of extensions of Lipschitz maps and of monotone maps, J
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Kirszbraun extensions preserving uniform distance in Hilbert spaces
A uniform-distance-preserving Kirszbraun extension exists iff the reference map satisfies a barycentric inequality, for arbitrary real Hilbert targets and subsets.