Constructing Optimal Maps for Monge’s Transport Problem as a Limit of Strictly Convex Costs

The Monge-Kantorovich problem is to move one distribution of mass onto another as efficiently as possible, where Monge’s original criterion for efficiency [19] was to minimize the average distance transported. Subsequently studied by many authors, it was not until 1976 that Sudakov showed solutions to be realized in the original sense of Monge, i.e., as mappings from R to R [23]. A second proof of this existence result formed the subject of a recent monograph by Evans and Gangbo [7], who avoided Sudakov’s measure decomposition results by using a partial differential equations approach. In the present manuscript, we give a third existence proof for optimal mappings, which has some advantages (and disadvantages) relative to existing approaches: it requires no continuity or separation of the mass distributions, yet our explicit construction yields more geometrical control than the abstract method of Sudakov. (Indeed, this control turns out to be essential for addressing a gap which has recently surfaced in Sudakov’s approach to the problem in dimensions n ≥ 3; see the remarks at the end of this section.) It is also shorter and more flexible than either, and can be adapted to handle transportation on Riemannian manifolds or around obstacles, as we plan to show in a subsequent work [13]. The problem considered here is the classical one: Problem 1 (Monge). Fix a norm d(x, y) = ‖x − y‖ on R, and two densities — non-negative Borel functions f, f− ∈ L(R) — satisfying the mass balance condition ∫