$p$-Wasserstein barycenters

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