A counterexample to the Monge ansatz for the three-marginal Coulomb cost

We construct a smooth compactly supported probability density on ℝ${}^3$, with a smooth compactly supported square root, for which the three-marginal Coulomb transport minimum is not attained by any pair of measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum. For every d ≥ 2 and s > 0, we also construct a smooth identical marginal with the same nonattainment and equality-of-infima properties for the inverse-power interaction ${\sum }_{i<j}|x_i-x_j{|}^{-s}$ on ℝ${}^d$.