p-Wasserstein barycenters

We study barycenters of $N$ probability measures on $R^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). 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 $p\ne 2$. 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.