  We introduce a general notion of transport cost that encompasses many costs
used in the literature (including the classical one and weak transport costs
introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type
duality theorem. As a by-product we obtain various applications in different
directions: we give a short proof of a result by Strassen on the existence of a
martingale with given marginals, we characterize the associated
transport-entropy inequalities together with the log-Sobolev inequality
restricted to convex/concave functions. Some explicit examples of discrete
measures satisfying weak transport-entropy inequalities are also given.
