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

It is known from clever mathematical examples [G. Carlier, Lecture Notes IMA, New Mathematical Models in Economics and Finance, (2010), pp. 1-82] that the Monge ansatz may fail in continuous 2-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, ℓ = 3 "sites," and symmetric pairwise costs with the values for N and ℓ both being optimal. Our counterexample is a transparent consequence of the convex geometry of the set of symmetric Kantorovich plans for N = ℓ = 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."