We introduce a new notion of (weak) transport cost on the Euclidean space that, with the help of the Kantorovich duality presented in the lecture by Paul-Marie Samson on Thursday, will give rise to some applications: (1) a new very short proof of a result by Strassen, (2) a characterization of those probablity measures satisfying the log-Sobolev inequality restricted to convex functions, and (3) a characterization of those probability measures on the line satisfying some Talagrand type transport-entropy inequality. Based on two joint works with Nathael Gozlan, Paul-Marie Samson, Yan Shu and Prasad Tetali.