Decomposition of the Kantorovich problem and Wasserstein distances on simplexes

Abstract

Let $X$ be a Polish space, $\mathcal{P}(X)$ be the set of Borel probability measures on $X$, and $T\colon X\to X$ be a homeomorphism. We prove that for the simplex $\mathrm{Dom} \subseteq \mathcal{P}(X)$ of all $T$-invariant measures, the Kantorovich metric on $\mathrm{Dom}$ can be reconstructed from its values on the set of extreme points. This fact is closely related to the following result: the invariant optimal transportation plan is a mixture of invariant optimal transportation plans between extreme points of the simplex. The latter result can be generalized to the case of the Kantorovich problem with additional linear constraints and the class of ergodic decomposable simplexes.