On the Core of a Low Dimensional Set-Valued Mapping

TL;DR

This paper provides new proofs for conditions ensuring the existence of Lipschitz selections for set-valued mappings in metric and Banach spaces, focusing on low-dimensional convex sets and utilizing a core reiteration formula.

Contribution

It offers alternative proofs for a Finiteness Principle for Lipschitz selections in special cases, simplifying the core analysis for low-dimensional set-valued mappings.

Findings

New proofs for the case m=2 in R^2

New proofs for the case m=1 in all Banach spaces

Introduction of a reiteration formula for the core of set-valued mappings

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 $f:{\mathcal M}\to X$ 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 $X={\bf R}^{2}$, 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 $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal M}$.