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Barycenters in the Wasserstein space. (English) Zbl 1223.49045

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.

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)

Citations:

Zbl 0889.49030
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