Before revision: We consider the optimal structure of couplings for the Martingale transport problem with initial and terminal laws \mu, \nu on R^d \R . Here optimality means that such solutions will minimize the functional E \E |X-Y|^p . We show how such optimal couplings are described and unique when 0<p  . We apply the monotonicity principle introduced in 4 , and the assumption of radial symmetry of \mu and \nu is important .
After revision: We consider the optimal structure of couplings for the Martingale transport problem with initial and terminal laws \mu, \nu on R^d \R^d and show the uniqueness of optimizer . Here optimality means that such solutions will minimize the functional E \E |X-Y|^p . We show how such optimal couplings are described and unique when 0<p  . We apply the monotonicity principle introduced in 4 , and the assumption of radial symmetry of \mu and \nu is important .