Agueh, Martial; Carlier, Guillaume Barycenters in the Wasserstein space. (English) Zbl 1223.49045 SIAM J. Math. Anal. 43, No. 2, 904-924 (2011). Summary: 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 W. Gangbo and A. Świȩch in [Commun. Pure Appl. Math. 51, No. 1, 23–45 (1998; Zbl 0889.49030)]. We also consider some examples and, in particular, rigorously solve the Gaussian case. We finally discuss convexity of functionals in the Wasserstein space. Cited in 5 ReviewsCited in 174 Documents MSC: 49Q20 Variational problems in a geometric measure-theoretic setting 49J40 Variational inequalities 49K21 Optimality conditions for problems involving relations other than differential equations 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49N15 Duality theory (optimization) Keywords:optimal transport; Wasserstein space; convexity; duality Citations:Zbl 0889.49030 PDFBibTeX XMLCite \textit{M. Agueh} and \textit{G. Carlier}, SIAM J. Math. Anal. 43, No. 2, 904--924 (2011; Zbl 1223.49045) Full Text: DOI HAL