Brunn-Minkowski type inequalities for the lattice point enumerator

TL;DR

This paper establishes new Brunn-Minkowski type inequalities for the lattice point enumerator, extending classical geometric inequalities to a discrete setting and exploring their implications and connections.

Contribution

The paper introduces novel discrete Brunn-Minkowski inequalities for lattice point counts, linking them to classical continuous inequalities and expanding the theoretical framework.

Findings

New inequalities for lattice point enumeration are proven.

Discrete inequalities imply classical Brunn-Minkowski results.

Connections with other geometric inequalities are discussed.

Abstract

Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are provided. In particular, we show that $$\mathrm{G}_n((1-\lambda)K + \lambda L + (-1,1)^n)^{1/n}\geq (1-\lambda)\mathrm{G}_n(K)^{1/n}+\lambda\mathrm{G}_n(L)^{1/n}$$ for any non-empty bounded sets $K, L\subset\mathbb{R}^n$ and all $\lambda\in(0,1)$. We also show that these new discrete versions imply the classical results, and discuss some links with other related inequalities.