Solution of the Monge problem II: Local approach

Abstract

In the previous chapter, we tried to establish the almost sure singlevaluedness of the c-subdifferential by an argument involving “global” topological properties, such as connectedness. Since this strategy worked out only in certain particular cases, we shall now explore a different method, based on local properties of c-convex functions. The idea is that the global question “Is the c-subdifferential of ψ at x single-valued or not?” might be much more subtle to attack than the local question “Is the function ψ differentiable at x or not?” For a large class of cost functions, these questions are in fact equivalent; but these different formulations suggest different strategies. So in this chapter, the emphasis will be on tangent vectors and gradients, rather than points in the c-subdifferential.