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It is known from clever mathematical examples \cite{Ca10} that the Mongeansatz may fail in continuous two-marginal optimal transport (alias optimalcoupling alias optimal assignment) problems. Here we show that this effectalready occurs for finite assignment problems with $N=3$ marginals, $\ell=3$'sites', and symmetric pairwise costs, with the values for $N$ and $\ell$ bothbeing optimal. Our counterexample is a transparent consequence of the convexgeometry of the set of symmetric Kantorovich plans for $N=\ell=3$, which -- aswe show -- possess 22 extreme points, only 7 of which are Monge. These extremepoints have a simple physical meaning as irreducible molecular packings, andthe example corresponds to finding the minimum energy packing forFrenkel-Kontorova interactions. Our finite example naturally gives rise, bysuperposition, to a continuous one, where failure of the Monge ansatz manifestsitself as nonattainment and formation of 'microstructure'.

A simple counterexample to the Monge ansatz in multi-marginal optimal  transport, convex geometry of the set of Kantorovich plans, and the  Frenkel-Kontorova model