p-Wasserstein barycenters

We study barycenters of N probability measures on R^d with respect to the p-Wasserstein metric (1<p<∞). We prove that – p-Wasserstein barycenters of absolutely continuous measures are unique, and again absolutely continuous – p-Wasserstein barycenters admit a multi-marginal formulation – the optimal multi-marginal plan is unique and of Monge form if the marginals are absolutely continuous, and its support has an explicit parametrization as a graph over any marginal space. This extends the Agueh–Carlier theory of Wasserstein barycenters [1] to exponents p≠2. A key ingredient is a quantitative injectivity estimate for the (highly non-injective) map from N-point configurations to their p-barycenter on the support of an optimal multi-marginal plan. We also discuss the statistical meaning of p-Wasserstein barycenters in one dimension.