Given a reductive group G and a parabolic subgroup P ⊂ G, with maximal torus T , we consider the closure X of a generic T-orbit (in the sense of Dabrowski's work), and determine when X is a Gorenstein-Fano variety. We establish a correspondence between the family of fans associated to a closure of a generic orbit and the family fans associated to a root system; these fans are characterized as those stable by the symmetries with respect to a facet. This correspondence is not bijective, but allows to determine which complete fans associated to a root system correspond to a Gorenstein-Fano variety. Lattice-regular convex polytopes arise as the polytopes associated to a sub-family of these fans — the lattice-regular complete fans.