  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 2 to several marginals, in the same sense that a well known
result of Gangbo and Swiech generalizes Brenier's polar factorization theorem
on $\mathbb{R}^n$.
