  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.
