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. (Lean)
Manuscript
A counterexample to the Monge ansatz for the three-marginal Coulomb cost
September 25, 2026 · source · cite as \cite{OAI:A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026}
We construct a smooth compactly supported probability density on ℝ3, 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 ℝd.
Result 373 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md