On the core of a low dimensional set-valued mapping

Abstract

Let $\mathfrak{M}=(\mathcal{M},\rho)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from $\mathcal{M}$ into the family $\mathcal{K}\_m(X)$ 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 $f\colon \mathcal{M}\to X$ such that $f(x)\in F(x)$ for every $x\in\mathcal{M}$. We give new alternative proofs of this result in two special cases. When $m=2$, we prove it for $X=\mathbb{R}^{2}$, 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 $G\colon \mathcal{M}\to\mathcal{K}\_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in\mathcal{M}$.