Resumen de On the core of a low dimensional set-valued mapping

Pavel Shvartsman

• Let M=(M,ρ) be a metric space and let X be a Banach space. Let F be a set-valued mapping from M 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 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 =2m=2, we prove it for 2X=R2, and when 1m=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