Lipschitz Partitions Of Unity Of Metric Spaces

A Lipschitz of unity of a metric space X is a set {Vj}j?J of Lipschitz functions from X to the unit interval[0,1]such that(1)for every x ? X,there is a neighborhood Ux of x where all but a finite number of the functions vj are zero;(2)the sum of all the function values at x is 1,namely,Ej?Jvj(x)= 1.For a Lipschitz partition of unity,in general,we have the additional requirement that it is subordinated to some open covering.Replacing Lipschitz functions by locally Lipschitz functions,we get the notion of locally Lipschitz partition of unity.In this paper,we first improve slightly Luukkainen and Vaisala's con-struction of locally Lipschitz functions,and give a detailed proof of the fol-lowing theorem stated by Luukkainen and Vaisala:For any open covering of a metric space,there is a locally Lipschitz partition of unity subordinated to this open cover.A direct corollary is that for any open covering of compact metric space,there is a Lipschitz partition of unity subordinated to the open cover.This corollary strengthen slightly Heinonen's a theorem.The second part of this paper concerns with the Lipschitz partition of unity subordinated to the whitney cover of an open set in Euclidean space.Heinonen pointed out such a Lipschitz partition of unity,without proof.In this paper,we explore some additional properties of Whitney cover,then give a detailed proof of this theorem.In this proof,we also give the precise repre-sentation of the corresponding constants appearing in the theorem.