Existence of Monge-type minimizers for multimarginal Coulomb optimal transport
Whether Monge-type minimizers always exist for the Coulomb cost is an open problem.
References
Primary source
Additional references
Source: 10.4171/OWR/2021/10, page 41. Canonicalized from 2 source-backed detection(s); richer provenance is retained in the corpus artifact.
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 1
RemarkAI-assistedClaimed by OpenAI. Claims a smooth compactly supported identical marginal in three-space whose three-marginal Coulomb minimum is unattainable by measure-preserving Borel maps although Monge and Kantorovich infima agree, with corresponding inverse-power examples in every dimension at least two.See full solution
Claimed by OpenAI. Claims a smooth compactly supported identical marginal in three-space whose three-marginal Coulomb minimum is unattainable by measure-preserving Borel maps although Monge and Kantorovich infima agree, with corresponding inverse-power examples in every dimension at least two.
Scope relative to this problem: The source reports a smooth compactly supported identical marginal in R^3 for which a three-marginal Coulomb minimum is not attained by measure-preserving Borel Monge maps. Equality of Monge and Kantorovich infima is preserved; this is failure of attainment, not an infimum gap. The source also reports inverse-power examples in dimensions at least two, without claiming failure of the two-marginal Coulomb Monge theorem.
GitHub repository: https://github.com/openai/math
- OpenAI-373-01-A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost.pdfOpen