Generalized Orlicz spaces and Wasserstein distances for convex–concave scale functions

Given a strictly increasing, continuous function ϑ:R+→R+ϑ:R+→R+, based on the cost functional∫X×Xϑ(d(x,y))dq(x,y), we define the LϑLϑ-Wasserstein distance Wϑ(μ,ν)Wϑ(μ,ν) between probability measures μ,νμ,ν on some metric space (X,d)(X,d). The function ϑ   will be assumed to admit a representation ϑ=φ∘ψϑ=φ∘ψ as a composition of a convex and a concave function φ and ψ  , resp. Besides convex functions and concave functions this includes all C2C2 functions.For such functions ϑ   we extend the concept of Orlicz spaces, defining the metric space Lϑ(X,m)Lϑ(X,m) of measurable functions f:X→Rf:X→R such that, for instance,dϑ(f,g)⩽1⇔∫Xϑ(|f(x)−g(x)|)dμ(x)⩽1.