Barycenters in the Wasserstein Space

TLDR

The paper introduces a generalized barycenter in the Wasserstein space extending McCann’s interpolation to more than two measures and discusses convexity of functionals in that space. The authors establish existence, uniqueness, characterization, and regularity of the barycenter and link it to the multimarginal optimal transport problem of Gangbo and Święch. They solve the Gaussian case rigorously and present additional examples. Citation: Pure Appl.

Abstract

In this paper, we introduce a notion of barycenter in the Wasserstein space which generalizes McCann's interpolation to the case of more than two measures. We provide existence, uniqueness, characterizations, and regularity of the barycenter and relate it to the multimarginal optimal transport problem considered by Gangbo and Święch in [Comm. Pure Appl. Math., 51 (1998), pp. 23–45]. We also consider some examples and, in particular, rigorously solve the Gaussian case. We finally discuss convexity of functionals in the Wasserstein space.