Projective reflection groups of finite covolume

We show that the Coxeter polytopes that have finite volume in their Vinberg domains are exactly the quasiperfect Coxeter polytopes of negative type, i.e. the Coxeter polytopes that are contained in their properly convex Vinberg domain, at the exception of some vertices that are C^1 points of the boundary. As a corollary, we show that for reflection groups à la Vinberg, the Vinberg domain is the only invariant properly convex domain if and only if the action is of finite covolume on the Vinberg domain and the dimension is at least 2.