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

TL;DR

This paper presents a finite counterexample demonstrating the failure of the Monge ansatz in multi-marginal optimal transport problems with three marginals and sites, linking convex geometry and physical models like Frenkel-Kontorova.

Contribution

It provides the first finite, explicit counterexample for three marginals, revealing geometric and physical insights into the Monge ansatz failure.

Findings

Counterexample with N=3, l=3, symmetric costs showing Monge ansatz failure

Convex geometry analysis reveals 22 extreme points, only 7 are Monge

Connection to physical models like molecular packings and Frenkel-Kontorova

Abstract

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'.