  We give a new definition of lattice-face polytopes by removing an unnecessary
restriction in the paper "Ehrhart polynomials of lattice-face polytopes", and
show that with the new definition, the Ehrhart polynomial of a lattice-face
polytope still has the property that each coefficient is the normalized volume
of a projection of the original polytope. Furthermore, we show that the new
family of lattice-face polytopes contains all possible combinatorial types of
rational polytopes.
