On existence of measure with given marginals supported on a hyperplane

Abstract

Let $\{{\mu }_k{\}}_{k=1}^N$ be absolutely continuous probability measures on the real line such that every measure ${\mu }_k$ is supported on the segment $[l_k,r_k]$ and the density function of ${\mu }_k$ is nonincreasing on that segment for all $k$. We prove that if $\mathbb{E}({\mu }_1)+\cdots +\mathbb{E}({\mu }_N)=C$ and if $r_k-l_k\le C-(l_1+\cdots +l_N)$ for all $k$, then there exists a transport plan with given marginals supported on the hyperplane $\{x_1+\cdots +x_N=C\}$. This transport plan is an optimal solution of the multimarginal Monge-Kantorovich problem for the repulsive harmonic cost function ${\sum }_{i,j=1}^N-(x_i-x_j{)}^2$.

28 pages, 1 figure