Optimal Transport Maps in Classical and Relativistic Geometries

In this talk I present a proof of the Monge problem of optimal transport which only relies on a non-branching property of geodesics and lower volume distortion assumptions. It is based on the proof of the Lp-Monge-Kantorovich problem, p>1, and applies to a wide range of geometries, in particular, to smooth Riemannian and Finsler manifolds as well as Lorentzian manifolds. Whereas previous proofs rely on the existence of dual solutions, the present proof for finite target measures is based on a simple geometric idea and its extension to arbitrary target measures is based on measure-theoretic arguments.