Nonattainment of the three-marginal Coulomb Monge problem

2 · Going further

The three-marginal problem prescribes the same probability distribution for each of three positions and minimizes their average interaction cost. In the Coulomb case on ${\mathbb{R}}^3$, the cost is ${\sum }_{i<j}|x_i-x_j{|}^{-1}$. The Monge formulation restricts configurations to $(x,T_2(x),T_3(x))$, where $T_2$ and $T_3$ are Borel maps preserving the prescribed distribution. The Kantorovich formulation allows general joint probability distributions with those same marginals.

The main manuscript constructs a smooth, compactly supported probability density on ${\mathbb{R}}^3$ whose square root is also smooth and compactly supported. For this density, it states that no pair of preserving Borel maps attains the Kantorovich minimum. At the same time, it constructs preserving maps whose costs approach that minimum, giving equality of the Monge and Kantorovich infima despite Monge nonattainment.

The abstract describes the argument at the level of these constructions: a density exhibiting nonattainment, together with preserving maps approaching the relaxed minimum. It does not supply the mechanism that excludes optimal maps or the details of the approximating construction. It also states the corresponding result for ${\sum }_{i<j}|x_i-x_j{|}^{-s}$ for every $d\ge 2$ and $s>0$, with a suitable smooth identical marginal in each case. Only the main manuscript is supplied here, so no relationship to companion manuscripts can be described.

Useful notions

• Probability densities and joint distributions
• Expected cost and minimization
• A minimum versus an infimum
• Maps that preserve a probability distribution
• Smoothness and compact support

What was known before / what is new

Limits and assumptions

These are existence results for specially constructed identical marginals, not claims about every smooth density. The Coulomb example is on three-dimensional space; the inverse-power extension assumes $d\ge 2$ and $s>0$. The abstract explicitly gives compact support and a smooth, compactly supported square root for the Coulomb density, but does not explicitly give those additional properties for every density in the extension. The main result has a Lean formalization; the supplied source does not specify which constructions or extensions that formalization covers. No remaining open problem is identified in the supplied text.

Mini glossary

Marginal

—

The probability distribution of one particle's position within a joint distribution of several positions.

Monge formulation

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The optimization problem in which the other particle positions are deterministic functions of the first, with the required distributions preserved.

Kantorovich formulation

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The optimization problem allowing any joint probability distribution with the prescribed marginals.

Infimum

—

The greatest lower bound on possible costs, whether or not any allowed configuration reaches it.

Nonattainment

—

The failure of every allowed candidate to reach the infimum.

Inverse-power interaction

—

An interaction whose pairwise cost is distance raised to a negative power, $|x_i-x_j{|}^{-s}$ with $s>0$.

Keywords : Optimal transport · Coulomb interaction · Monge nonattainment

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family