2014/12/24 by Young‐Heon Kim, Kim, Young-Heon, Brendan Pass +1 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1412.7726
openalex publication_date 2014/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω∈ P(P(M)) on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier \citeac to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger \citec-ems for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn-Minkowski inequality for a random measurable set on a Riemannian manifold.