Before revision:  We show that such probability measure exists if and only if the \alpha average of the so-called G-expectation of bounded continuous functions with respect to the measures in \Gamma is less than their \beta average. 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.
After revision:  First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem with constraints and obtain the Kantorovich duality.