On the core of a low dimensional set-valued mapping

Abstract

LetM = (M, ρ) be a metric space and let X be a Banach space. Let F be a setvalued mapping fromM into the family Km(X) of all compact convex subsets of X of dimension at most m. The main result in our recent joint paper [16] with Charles Fefferman (which is referred to as a “Finiteness Principle for Lipschitz selections”) provides efficient conditions for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f :M→ X such that f (x) ∈ F(x) for every x ∈ M. We give new alternative proofs of this result in two special cases. When m = 2 we prove it for X = R, and when m = 1 we prove it for all choices of X. Both of these proofs make use of a simple reiteration formula for the “core” of a set-valued mapping F, i.e., for a mapping G :M→ Km(X) which is Lipschitz with respect to the Hausdorff distance, and such that G(x) ⊂ F(x) for all x ∈ M.