(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 $N\subseteq M$ there always exists an almost isometric local retract $S\subseteq M$ with $N\subseteq S$ and $dens\left(N\right)=dens\left(S\right)$. 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.