The multistochastic Monge-Kantorovich problem

Abstract

The multistsochastic Monge--Kantorovich problem on the product $X={\prod }_{i=1}^n{X}_i$ of $n$ spaces is a generalization of the multimarginal Monge--Kantorovich problem. For a given integer number $1\le k<n$ we consider the minimization problem $\int cd\pi \to \inf$ of the space of measures with fixed projections onto every $X_{i_1}\times \cdots \times X_{i_k}$ for arbitrary set of $k$ indices $\{i_1,\dots ,i_k\}\subset \{1,\dots ,n\}$. In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual solution.

70 pages, 6 figures