American Journal of Mathematics

We study a multi-marginal optimal transportation problem on a Riemannian manifold, with cost function given by the average distance squared from multiple points to their barycenter. Under a standard regularity condition on the first marginal, we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable, thus inducing a Monge solution. This result generalizes McCann's polar factorization theorem on manifolds from two to several marginals, in the same sense that a well-known result of Gangbo and Swiech generalizes Brenier's polar factorization theorem on ${\Bbb R}^n$.