Disintegrated optimal transport for metric fiber bundles

Abstract: We define a new two-parameter family of metrics on a subset of Borel probability measures on a general metric fiber bundle, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family of metrics contains the classical Monge-Kantorovich metrics, linearized optimal transport distance, and generalizes the sliced and max-sliced Wasserstein metrics. We prove these metrics are complete, separable (except an endpoint case), geodesic spaces, with a dual representation. Additionally, we prove existence and duality for an associated barycenter problem, and provide conditions for uniqueness of the barycenter. These results on barycenter problems for the disintegrated Monge--Kantorovich metrics also yield the corresponding existence, duality, and uniqueness results for classical Monge--Kantorovich barycenters in a wide variety of spaces, including a uniqueness result on any connected, complete Riemannian manifold, with or without boundary; this is the first and only result with absolutely no restriction on the geometry of the manifold (such as on curvatures or injectivity radii). Our results cannot be obtained by applying the theory of $L^q$ maps valued in spaces of probability measures, in fact the $L^q$ map case can be recovered from our results by taking the underlying bundle as a trivial product bundle.