Lipschitz Neighborhood Retraction And Its Applications

A subset A of a metric space X is a Lipschitz neighborhood retract if there is a neighborhood of A in X and a Lipschitz map from U to A such that the restriction of r on A is the identity map of A.When X is a Euclidean space,one has the notion of Lipschitz Euclidean neighborhood retract.In this paper,we first give a Lipschitz version of the intrinsic property of(topologi?cal)Euclidean neighborhood retract.In addition,we give a detailed proof of Almgren'5 a theorem concerned with Lipschitz neighborhood retract,and im?prove the corresponding constants.Finally,we obtain a positive result about the following problem:Under what conditions,a subset of Euclidean space is contractible.