$p$-Wasserstein barycenters

We study barycenters of $N$ probability measures on $\mathbb{R}^d$ 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 [SIAM J. Math. Anal. 43 (2011), no.2, 904--924] to exponents $p\neq 2$. 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.