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

Given a strictly increasing, continuous function $\vartheta:\R_+\to\R_+$, based on the cost functional $\int_{X\times X}\vartheta(d(x,y))\,d q(x,y)$, we define the $L^\vartheta$-Wasserstein distance $W_\vartheta(\mu,\nu)$ between probability measures $\mu,\nu$ on some metric space $(X,d)$. The function $\vartheta$ will be assumed to admit a representation $\vartheta=\phi\circ\psi$ as a composition of a convex and a concave function $\phi$ and $\psi$, resp. Besides convex functions and concave functions this includes all $\mathcal C^2$ functions. For such functions $\vartheta$ we extend the concept of Orlicz spaces, defining the metric space $L^\vartheta(X,m)$ of measurable functions $f: X\to\R$ such that, for instance, $$d_\vartheta(f,g)\le1\quad\Longleftrightarrow\quad \int_X\vartheta(|f(x)-g(x)|)\,d\mu(x)\le1.$$