There is a smooth compactly supported probability density on ℝ³, with smooth compactly supported square root, whose finite three-marginal Coulomb Kantorovich minimum is attained by a coupling but by no pair of measure-preserving Borel maps; nevertheless the Monge and Kantorovich infima are equal.
There is a smooth compactly supported probability density on ℝ³, with smooth compactly supported square root, whose finite three-marginal Coulomb Kantorovich minimum is attained by a coupling but by no pair of measure-preserving Borel maps; nevertheless the Monge and Kantorovich infima are equal.
There is a smooth compactly supported probability density on ℝ³, with smooth compactly supported square root, whose finite three-marginal Coulomb Kantorovich minimum is attained by a coupling but by no pair of measure-preserving Borel maps; nevertheless the Monge and Kantorovich infima are equal.
If correct, gives a smooth physical-density counterexample to attainment of the Coulomb co-motion ansatz.
Must a three-marginal Coulomb optimum for a smooth identical density be attained by measure-preserving maps?
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OpenAI reports nonattainment of the three-marginal Coulomb Monge minimum
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