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

siam journal on mathematical analysis

A Simple Counterexample to the Monge Ansatz in Multimarginal Optimal Transport, Convex Geometry of the Set of Kantorovich Plans, and the Frenkel--Kontorova Model