An explicit solution for a multimarginal mass transportation problem

TL;DR

This paper provides an explicit solution to a specific multimarginal mass transportation problem with a cubic cost function on the unit cube, revealing unique dual solutions and complex primal solution structures.

Contribution

It introduces an explicit solution for a multimarginal transportation problem with a particular cost function and analyzes the structure of primal and dual solutions.

Findings

Primal solutions are concentrated on a set with non-constant local dimension.

The dual problem has a unique solution up to additive constants.

The problem demonstrates complex solution structures in multimarginal mass transportation.

Abstract

We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0,1]^3$ with the cost function $xyz$ and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with non-constant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique (up to addition of constants).