Sharp finiteness principles for Lipschitz selections: long version

Abstract

Let $(\mathcal{M},\ddot{I})$ 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.

79 pages, 2 figures