  We study finite energy classes of quasiplurisubharmonic (qpsh) functions in
the setting of toric compact K{\"a}hler manifolds. We characterize toric qpsh
functions and give necessary and sufficient conditions for them to have finite
(weighted) energy, both in terms of the associated convex function in R n , 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 qpsh 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 H{\"o}lder
continuous toric quasisubharmonic functions.
