  We study inequalities that simultaneously relate the number of lattice
points, the volume and the successive minima of a convex body to one another.
One main ingredient in order to establish these relations is Blaschke's shaking
procedure, by which the problem can be reduced from arbitrary convex bodies to
anti-blocking bodies. As a consequence of our results, we obtain an upper bound
on the lattice point enumerator in terms of the successive minima, which is
equivalent to Minkowski's upper bound on the volume in terms of the successive
minima.
