Inserted: 1 oct 2012
Last Updated: 1 oct 2012
We consider optimal transport problems in Rd as well as on manifolds for cost functions c which satisfy a nonsmooth version of the classical Left Twist condition (i.e. the invertibility of the gradient in the first variable). Under the classical assumption that the initial measure does not give mass to sets with σ-finite Hd−1 measure, we provide a short and self-contained proof of the fact that any optimal transport plan as well as any feasible transport plan satisfying a c-monotonicity assumption is induced by a transport map. We also show that the usual costs induced by Tonelli Lagrangians satisfy the Nonsmooth Left Twist condition we propose.
Keywords: Monge-Kantorovich problem, optimal transport problem, cyclical monotonicity, Tonelli Lagrangian