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A natural and important question in multi-marginal optimal transport iswhether the \emph{Monge ansatz} is justified; does there exist a solution ofMonge, or deterministic, form? We address this question for the quadratic costwhen each marginal measure is $m$-empirical (that is, uniformly supported on$m$ points). By direct computation, we provide an example showing that theansatz \emph{can fail} when the underlying dimension $d$ is $2$, number ofmarginals $N$ 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 barycentricallyconvex when $N \geq 3$, $d \geq 2$ and $m \geq3$. It is a well knownconsequence 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 simpleproof here that \emph{it holds whenever $m=2$}. Therefore, the $N$, $d$ and $m$in our counterexample are as small as possible.

On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals