On Rockafellar’s theorem

In this talk, we will describe the problem of finding an optimal transport map (with respect to a cost), and Rockafellar’s theorem which is a main component in solving this problem. We will recall Rockafellar’s elegant proof, and see why it fails when the cost is allowed to attain the value infinite. We will give a new proof which does work in this case, utilizing a new notion of c-path-boundedness, which replaces that of c-cyclic monotonicity in the classical theory.