Calculus of Variations

Monge’s problem refers to the classical problem of optimally transporting mass: given Borel probability measures μ = μ− on R, find the measure preserving map s(x) between them which minimizes the average distance transported. Here distance can be induced by the Euclidean norm, or any other uniformly convex and smooth norm d(x, y) = ‖x− y‖ onR. Although the solution is never unique, we give a geometrical monotonicity condition singling out a particular optimal map s(x). Furthermore, a local definition is given for the transport cost density associated to each optimal map. All optimal maps are then shown to lead to the same transport density a ∈ L(R). Mathematics Subject Classification (2000): 49Q20, 28A50