Barycenters in Disintegrated optimal transport

TL;DR

This paper establishes existence, duality, and uniqueness results for Monge--Kantorovich barycenters on a broad class of metric spaces, including all connected, complete Riemannian manifolds, using a novel family of metrics on probability measures.

Contribution

It introduces disintegrated Monge--Kantorovich metrics and proves the first general uniqueness results for barycenters on arbitrary connected, complete Riemannian manifolds.

Findings

Proved existence and duality for barycenters in broad metric spaces.

Established uniqueness of barycenters on connected, complete Riemannian manifolds.

Developed a new two-parameter family of metrics on probability measures.

Abstract

We prove existence and duality on a wide class of metric spaces, and uniqueness results on any connected, complete Riemannian manifold, with or without boundary, for classical Monge--Kantorovich barycenters. In particular, this is the first and only uniqueness result with no restriction on the geometry of the manifold aside from connectedness and completeness. We obtain these via the corresponding results for barycenter problems associated to a new two-parameter family of metrics on probability measures on a general metric fiber bundle, called the $\textit{disintegrated Monge--Kantorovich metrics}$ (previously introduced by the authors).