  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.
