  A natural and important question in multi-marginal optimal transport is
whether the \emph{Monge ansatz} is justified; does there exist a solution of
Monge, or deterministic, form? We address this question for the quadratic cost
when each marginal measure is $m$-empirical (that is, uniformly supported on
$m$ points). By direct computation, we provide an example showing that the
ansatz \emph{can fail} when the underlying dimension $d$ is $2$, number of
marginals $N$ to be matched is $3$ and the size $m$ of their supports is $3$.
As a consequence, the set of $m$-empirical measures is not barycentrically
convex when $N \geq 3$, $d \geq 2$ and $m \geq3$. It is a well known
consequence of the Birkhoff-von Neumann Theorem that the Monge ansatz holds for
$N=2$, standard techniques show it holds when $d=1$, and we provide a simple
proof here that \emph{it holds whenever $m=2$}. Therefore, the $N$, $d$ and $m$
in our counterexample are as small as possible.
