Before revision:  The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of  X and the dual problem is defined on the bidual of X . These results are then applied to several extensions of the classical optimal transport .
After revision:  The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of  X and the dual problem is defined on the  X . These results are then applied to several extensions of the classical optimal transport .