2024/02/20 by Brizzi, Camilla, Friesecke, Gero, Ried, Tobias · 1 citation
#49K21 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 49Q20 #Secondary 49J40
paper · doi:10.48550/arxiv.2402.13176
We generalize the notion and theory of Wasserstein barycenters introduced by Agueh and Carlier (2011) from the quadratic cost to general smooth strictly convex costs h with non-degenerate Hessian. We show the equivalence between a coupled two-marginal and a multi-marginal formulation and establish that the multi-marginal optimal plan is unique and of Monge form. To establish the latter result we introduce a new approach which is not based on explicitly solving the optimality system, but instead deriving a quantitative injectivity estimate for the (highly non-injective) map from N-point configurations to their h-barycenter on the support of an optimal multi-marginal plan.