(Almost isometric) local retracts in metric spaces

Abstract

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of ideals and almost isometric ideals in Banach spaces. We prove that given two metric spaces N⊆Mthere always exists an almost isometric local retract S⊆Mwith N⊆Sand dens(N) =dens(S). We also prove that metric spaces which are local retracts (respectively almost isometric local retracts) can be characterised in terms of a condition of extendability of Lipschitz functions (respectively almost isometries) between finite metric spaces. Various examples and counterexamples are exhibited.