  Let ${\mathfrak M}=({\mathcal M},\rho)$ be a metric space and let $X$ be a
Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the
family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at
most $m$. The main result in our recent joint paper with Charles Fefferman
(which is referred to as a ``Finiteness Principle for Lipschitz selections'')
provides efficient conditions for the existence of a Lipschitz selection of
$F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$
for every $x\in{\mathcal M}$. We give new alternative proofs of this result in
two special cases. When $m=2$ we prove it for $X={\bf R}^{2}$, and when $m=1$
we prove it for all choices of $X$. Both of these proofs make use of a simple
reiteration formula for the ``core'' of a set-valued mapping $F$, i.e., for a
mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to
the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal
M}$.
