1. An Explicit Solution for a Multimarginal Mass Transportation Problem
- Author
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Nikita A. Gladkov and Alexander P. Zimin
- Subjects
Applied Mathematics ,Mathematical analysis ,Construct (python library) ,Transportation theory ,Function (mathematics) ,Computational Mathematics ,Optimization and Control (math.OC) ,Unit cube ,FOS: Mathematics ,Applied mathematics ,Mass transportation ,Mathematics - Optimization and Control ,Analysis ,Mathematics - 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)., Comment: 31 pages, 4 figures. The paper was completely rewritten. Heuristic considerations to find a solution of the primal problem added. Algorithm to find the primal problem solution numerically added (arbitrary marginals). The construction was generalized for a C(ln x + ln y + ln z), C is convex. Measure on the triangle was found with the support singular with respect to the Lebesgue measure
- Published
- 2020
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