  Gardner, Gronchi and Zong posed the problem to find a discrete analogue of M.
Meyer's inequality bounding the volume of a convex body from below by the
geometric mean of the volumes of its slices with the coordinate hyperplanes.
Motivated by this problem, for which we provide a first general bound, we study
in a more general context the question to bound the number of lattice points of
a convex body in terms of slices as well as projections.
