{"id":"openai-math/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost","pdf":"https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026","tags":["optimal-transport","coulomb-cost","multi-marginal","monge-problem"],"score":0,"title":"A counterexample to the Monge ansatz for the three-marginal Coulomb cost","domain":"Partial differential equations","family":"373","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AP","summary":"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 .","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"Nonattainment of the three-marginal Coulomb Monge problem"}