On Rogers–Shephard-type inequalities for the lattice point enumerator

Abstract

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.