Preprint
Inserted: 29 sep 2024
Last Updated: 29 sep 2024
Year: 2024
Notes:
This is the first paper in our series of papers on Wasserstein barycenter problem. All questions and comments are welcome.
Abstract:
We study the Wasserstein barycenter problem in the setting of infinite-dimensional, non-proper, non-smooth extended metric measure spaces. We introduce a couple of new concepts and study the existence, uniqueness, absolute continuity of the 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, some extended metric measure spaces including abstract Wiener spaces.
We also introduce a relaxation of the CD condition, we call the Barycenter-Curvature-Dimension condition BCD. 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 inequalities including a multi-marginal Brunn--Minkowski inequality and a functional Blaschke--Santalo type inequality.
Keywords: Optimal transport, Ricci curvature, metric measure space, Wasserstein barycenter, curvature-dimension condition
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