MSI Weekly Bulletin - Week starting Monday 22 May, 2006

The problem of optimal transportation is the following: given two probabililty measures, find a map that transports one onto the other, minimizing the integral of a cost function which depends only on the departure and arrival points. Although this is an old problem that goes back to Monge, it has come back to light since the discovery of Brenier in the mid eighties who found that when the cost is quadratic, optimal maps are just gradient of convex functions. A general theory is now available concerning the existence of the minimizers for general cost functions, with a generalization of the concept of convexity to c-convexity. All optimal maps derive from a c-convex potential, in turn solution to an elliptic Monge-Ampere equation. We will develop on the notion of c-convexity, and explain how a natural geometric condition on the cost function can been shown to be also a necessary and sufficient condition for smoothness of solutions of the associated Monge-Ampere equation when the measures are smooth. This "necessary" part thus completes the "sufficient" part recently obtained by Ma Trudinger and Wang. Finally we will discuss how this condition can be related to the curvature of the underlying manifold.