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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.

In this paper we study various Rogers-Shephard type inequalities for the lattice point enumerator Gn(•) on R n . In particular, for any non-empty convex bounded sets K, L ⊂ R n , we show that

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In this paper, we study various Rogers–Shephard-type inequalities for the lattice point enumerator [Formula: see text] on [Formula: see text]. In particular, for any non-empty convex bounded sets [Formula: see text], we show that [Formula: see text] and [Formula: see text] for [Formula: see text], [Formula: see text]. Additionally, a discrete counterpart to a classical result by Berwald for concave functions, from which other discrete Rogers–Shephard-type inequalities may be derived, is shown. Furthermore, we prove that these new discrete analogues for [Formula: see text] imply the corresponding results involving the Lebesgue measure.