Toric pluripotential theory

We study finite energy classes of quasiplurisubharmonic functions in the setting of toric compact Kähler manifolds. We characterize toric quasiplurisubharmonic functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in Rⁿ, and through the integrability properties of its Legendre transform. We characterize log-Lipschitz convex functions on the Delzant polytope, showing that they correspond to toric quasiplurisubharmonic functions which satisfy a certain exponential integrability condition. In the particular case of dimension one, those log-Lipschitz convex functions of the polytope correspond to Hölder continuous toric quasisubharmonic functions.