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 ${\mathrm{G}}_n(\cdot )$ on ${\mathbb{R}}^n$. In particular, for any non-empty convex bounded sets $K,L\subset {\mathbb{R}}^n$, we show that \[\mathrm{G}_{n}(K+L)\mathrm{G}_{n}\bigl(K\cap(-L)\bigr) \leq\binom{2n}{n} \mathrm{G}_{n}\bigl(K+(-1,1)^n\bigr)\mathrm{G}_{n}\bigl(L+(-1,1)^n\bigr). \] and \[ \mathrm{G}_{n-k}(P_{H^\perp} K)\mathrm{G}_{k}(K\cap H)\leq\binom{n}{k}\mathrm{G}_{n}\bigl(K+(-1,1)^n\bigr), \] for $H=\mathrm{lin}\{{\mathrm{e}}_1,\dots ,{\mathrm{e}}_k\}$, $k\in \{1,\dots ,n-1\}$. 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 ${\mathrm{G}}_n(\cdot )$ imply the corresponding results involving the Lebesgue measure.