Sharp finiteness principles for Lipschitz selections

TL;DR

This paper establishes a precise finiteness principle for the existence of Lipschitz selections of set-valued mappings from metric spaces into convex subsets of Banach spaces, with optimal constants.

Contribution

It introduces a sharp finiteness principle for Lipschitz selections of set-valued maps with convex compact values of bounded dimension.

Findings

Proves a finiteness principle with the sharp constant for Lipschitz selections.

Extends the theory of Lipschitz selections to convex compact sets of bounded dimension.

Provides a criterion for the existence of Lipschitz selections based on finite subsets.

Abstract

Let $(M,\rho)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from $M$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of $F$ with the sharp value of the finiteness constant.