On Rogers-Shephard type inequalities for the lattice point enumerator

TL;DR

This paper establishes new Rogers-Shephard type inequalities for the lattice point enumerator in convex geometry, providing discrete analogues of classical continuous results and linking them to Lebesgue measure inequalities.

Contribution

It introduces novel discrete Rogers-Shephard inequalities for lattice point counts and connects these to classical measure-based inequalities, extending the theory to discrete settings.

Findings

Derived inequalities for lattice point enumerator involving Minkowski sums and intersections.

Established a discrete analogue to Berwald's classical result for concave functions.

Linked discrete inequalities to continuous measure inequalities, broadening their applicability.

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.