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.