Before revision:  As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times  .
After revision:  As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times . Second, we consider the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle which gives us a geometric way of identifying the optimizer .