  Minkowski's second theorem on successive minima gives an upper bound on the
volume of a convex body in terms of its successive minima. We study the problem
to generalize Minkowski's bound by replacing the volume by the lattice point
enumerator of a convex body. In this context we are interested in bounds on the
coefficients of Ehrhart polynomials of lattice polytopes via the successive
minima. Our results for lattice zonotopes and lattice-face polytopes imply, in
particular, that for 0-symmetric lattice-face polytopes and lattice
parallelepipeds the volume can be replaced by the lattice point enumerator.
