Partial Regularity Results in Optimal Transportation

This note describes some recent results on the regularity of optimal transport maps. As we shall see, in general optimal maps are not globally smooth, but they are so outside a “singular set” of measure zero. 1. The optimal transportation problem The optimal transportation problem, whose origin dates back to Monge [19], aims to find a way to transport a distribution of mass from one place to another by minimizing the transportation cost. Mathematically, the problem can be formulated as follows: given two probability measures μ and ν (representing respectively the initial and final configuration of the mass that we want to transport) defined on the measurable spaces X and Y , one says that a map T : X → Y transports μ onto ν if T]μ = ν, i.e., ν(A) = μ ( T−1(A) ) ∀A ⊂ Y measurable. Then, given a cost function c : X×Y → R (so that c(x, y) represents the cost to transport a unit of mass from x to y), one wants to minimize the transportation cost among all possible transport maps. Since transporting a unitary mass from x to T (x) costs c(x, T (x)), the cost to transport the whole mass μ is simply given by ∫ X c(x, T (x)) dμ(x). Hence the optimal transportation problem consists in solving the minimization problem