Local semiconvexity of Kantorovich potentials on non-compact manifolds

In the theory of optimal transportation, one wants to pair sources with sinks, so as to minimize the average cost c(x,y) of transportation. Historically, there were two approaches to doing so: Monge's (1781) was to assign each source to a single sink, whereas Kantorovich's (1942) allowed the freedom of dividing the output of each source between many sinks. The latter problem is a linear program -- infinite-dimensional when sources or sinks are spread continuously over the landscape. It is solved through a dual linear program which assigns implicit values to the goods being transported, depending on the merits of their location. These values are called Kantorovich potentials; when they exist, their spatial derivatives often contain enough information to allow the problem of Monge to be solved also.

The contribution of Figalli and Gigli is to show such potentials have two derivatives almost everywhere -- the same regularity as concave functions. In fact, they differ from a concave function by a smooth function. For compactly supported measures and/or special cost functions, such results go back to Brenier (1987) and Gangbo and McCann (1996); they are key to solving variants of Monge's problem, and the Monge-Ampère equation. Figalli and Gigli show this extends to measures with non-compact support, assuming only mild growth conditions on the cost c in C2. This is a deep result with multifold applications, as examples include a wide variety of cost functions arising from Tonelli-Lagrangians, such as Riemannian distance squared costs, and natural mechanical actions.

Irène Fonseca
Corresponding Editor

Local semiconvexity of Kantorovich potentials on non-compact manifolds
Alessio Figalli and Nicola Gigli
ESAIM: COCV 17 (2011) 648-653