Barycenters in the Wasserstein space

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 'Swi¸ech in [8]. We also consider some examples and in particular rigorously solve the gaussian case. We finally discuss convexity of functionals in the Wasserstein space.ou