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 $\mathbb{E}(\mu_1) + \dots + \mathbb{E}(\mu_N) = C$ and if $r_k - l_k \le C - (l_1 + \dots + l_N)$ for all $k$, then there exists a transport plan with given marginals supported on the hyperplane $\{x_1 + \dots + x_N = C\}$. This transport plan is an optimal solution of the multimarginal Monge-Kantorovich problem for the repulsive harmonic cost function $\sum_{i, j = 1}^N-(x_i - x_j)^2$.