论文标题
在存在衡量标准的情况下，在超平面上支撑的边缘
On existence of measure with given marginals supported on a hyperplane
论文作者
论文摘要
令$ \ {μ_k\} _ {k = 1}^n $绝对是实际行上的连续概率度量，以便每种度量$μ_k$都在段$ [l_k，r_k] $上支持$μ_k$的密度函数，并且对所有$ k $ a sembering noncransing noncressing。我们证明，如果$ \ mathbb {e}（μ_1） + \ dots + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \μ_n）= c $，如果$ r_k -l_k -l_k \ le c \ le c - （l_1 + \ dots + dots + dots + l_n）$ in $ k $，则与给定的Marginals Supports + flans in Marginals Suppertanne + \ d + \ x_n = c \} $。该运输计划是用于排斥谐波成本功能$ \ sum_ {i，j = 1}^n-（x_i-x_j）^2 $的多边形蒙格 - 坎托维奇问题的最佳解决方案。
Let $\{μ_k\}_{k = 1}^N$ be absolutely continuous probability measures on the real line such that every measure $μ_k$ is supported on the segment $[l_k, r_k]$ and the density function of $μ_k$ is nonincreasing on that segment for all $k$. We prove that if $\mathbb{E}(μ_1) + \dots + \mathbb{E}(μ_N) = C$ 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 $\sum_{i, j = 1}^N-(x_i - x_j)^2$.
