  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.
