Uniqueness of optimal plans for multi-marginal mass transport problems via a reduction argument

Abstract

For a family of probability spaces $\{(X_k,\mathcal{B}_{X_k},\mu_k)\}_{k=1}^N$ and a cost function $c: X_1\times\cdots\times X_N\to \mathbb{R}$ we consider the Monge-Kantorovich problem \begin{align*}\tag{MK}\label{MONKANT} \inf_{\lambda\in\Pi(\mu_1,\ldots,\mu_N)}\int_{\prod_{k=1}^N X_k}c\,d\lambda. \end{align*} Then for each ordered subset $\mathcal{P}=\{i_1,\ldots,i_p\}\subsetneq\{1,...,N\}$ with $p\geq 2$ we create a new cost function $c_\mathcal{P}$ corresponding to the original cost function $c$ defined on $\prod_{k=1}^p X_{i_k}$. This new cost function $c_\mathcal{P}$ enjoys many of the features of the original cost $c$ while it has the property that any optimal plan $\lambda$ of \eqref{MONKANT} restricted to $\prod_{k=1}^p X_{i_k}$ is also an optimal plan to the problem \begin{align*}\tag{RMK}\label{REDMONKANT} \inf_{\tau\in\Pi(\mu_{i_1},\ldots\mu_{i_p})}\int_{\prod_{k=1}^p X_{i_k}}c_{\mathcal{P}}\,d\tau. \end{align*} Our main contribution in this paper is to show that, for appropriate choices of index set $\mathcal{P}$, one can recover the optimal plans of \eqref{MONKANT} from \eqref{REDMONKANT}. In particular, we study situations in which the problem \eqref{MONKANT} admits a unique solution depending on the uniqueness of the solution for the lower marginal problems of the form \eqref{REDMONKANT}. This allows us to prove many uniqueness results for multi-marginal problems when the unique optimal plan is not necessarily induced by a map. To this end, we extensively benefit from disintegration theorems and the $c$-extremality notions. Moreover, by employing this argument, besides recovering many standard results on the subject including the pioneering work of Gangbo-\'Swi\c ech, several new applications will be demonstrated to evince the applicability of this argument.