On the geometry of Wasserstein barycenter I

We study the Wasserstein barycenter problem in the setting of non-compact, non-smooth extended metric measure spaces. We introduce a couple of new concepts and obtain the existence, uniqueness, absolute continuity of the Wasserstein barycenter, and prove Jensen's inequality in an abstract framework. This generalized several results on Euclidean space, Riemannian manifolds and Alexandrov spaces, to metric measure spaces satisfying Riemannian Curvature-Dimension condition \`a la Lott--Sturm--Villani, and some extended metric measure spaces including abstract Wiener spaces and configuration spaces over Riemannian manifolds. We also introduce a new curvature-dimesion condition, we call Barycenter-Curvature-Dimension condition. We prove its stability under measured-Gromov--Hausdorff convergence and prove the existence of the Wasserstein barycenter under this new condition. In addition, we get some geometric inequalities including a multi-marginal Brunn--Minkowski inequality and a functional Blaschke--Santal\'o type inequality.