Continuity of extensions of Lipschitz maps and of monotone maps

Abstract

Let be a subset of a Hilbert space. We prove that if is such that for all and all non‐negative that add up to one, then for any 1‐Lipschitz , with , there exists a 1‐Lipschitz extension of such that the uniform distance on between and is the same as the uniform distance on between and .Moreover, if either or is convex, we prove the converse: We show that a map that allows for a 1‐Lipschitz, uniform distance preserving extension of any 1‐Lipschitz map on a subset of 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.