Continuity of extensions of Lipschitz maps and of monotone maps

Abstract

Let $X$ be a subset of a Hilbert space. We prove that if $v:X\to R^m$ is such that

Moreover, if either $m\in \{1,2,3\}$ or $X$ is convex, we prove the converse: We show that a map $v:X\to R^m$ that allows for a 1-Lipschitz, uniform distance preserving extension of any 1-Lipschitz map on a subset of $X$ also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.