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

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OpenAI Mathopenai-math/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-costSep 25, 2026math.APAI-generatedscore 0

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 $\sum_{i\lt j}|x_i-x_j|^{-s}$ on ℝ d .

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