  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.
