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Working paper

Decomposition of the Kantorovich problem and Wasserstein distances on simplexes

Zaev D.
We consider L^p-Wasserstein distances on a subset of probability measures. If the subset of interest appears to be a simplex, these distances are determined by their values on extreme points of the simplex. We show that this fact is a corollary of the following decomposition result: an optimal transport plan can be represented as a mixture of optimal transport plans between extreme points of the simplex. This fact can be generalized to the Kantorovich problem with additional linear restriction and the associated Wasserstein-like distances. We prove that the decomposition is possible, if marginal measures are elements of a simplex that is compatible with the additional restriction.