Sharp finiteness principles for Lipschitz selections: long version

TL;DR

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

Contribution

It introduces a new finiteness principle with the optimal finiteness number for Lipschitz selections in metric and Banach space settings.

Findings

Proves a sharp finiteness principle for Lipschitz selections.

Determines the exact finiteness number needed for existence.

Extends previous results to convex compact subsets of Banach spaces.

Abstract

Let $({\mathcal M},\rho)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from ${\mathcal M}$ 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 number.