Nonattainment of the three-marginal Coulomb Monge problem
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.
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MANUSCRIPT 01
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…
Full source abstract
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).
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