On the structure of optimal martingale transport plans in general dimensions

I will describe the profile of optimal solutions of the martingale counterpart of the Monge mass transport problem. These are one-step martingales that maximize or minimize the expected value of the modulus of their increment among all martingales having two prescribed —convex ordered-- probability measures as marginals. While there is a great deal of results —mostly established by the mathematical financial community— when the marginals are probabilities on the real line, much less is known in the richer and more delicate higher dimensional setting.