Barycenters in the Wasserstein space

Martial Agueh (1) , Guillaume Carlier (2)

1 DMS - Department of Mathematics and Statistics
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Domaines: Optimisation et contrôle [math.OC]

Martial Agueh, Guillaume Carlier. Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 2011, 43 (2), pp.904-924. ⟨10.1137/100805741⟩. ⟨hal-00637399⟩