h-Wasserstein barycenters

We generalize the notion and theory of Wasserstein barycenters [1] from the quadratic cost to general smooth strictly convex costs h with non-degenerate Hessian. We show the equivalence between a coupled two-marginal and a multi-marginal formulation and establish that the multi-marginal optimal plan is unique and of Monge form. The latter result is a special case of deep abstract results by Pass [14] relying on the machinery of geometric measure theory. Here we provide a new, more quantitative and we believe simpler proof, by deriving an injectivity estimate for the (highly non-injective) map from N -point configurations to their h -barycenter on the support of an optimal multi-marginal plan.