Barycenters in the wasserstein space

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 'Swiȩch 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. © 2011 Society for Industrial and Applied Mathematics.