Lattice polytopes with the minimal volume

TL;DR

This paper investigates lattice polytopes with minimal volume, characterizing classes where the lower bound on volume, derived from Ehrhart polynomial bounds, is exactly attained, especially in relation to boundary and interior lattice points.

Contribution

The paper identifies and describes classes of lattice polytopes that achieve the minimal volume bound given by Ehrhart polynomial inequalities, extending known results in specific dimensions.

Findings

Characterization of lattice polytopes with minimal volume

Identification of classes where the Ehrhart bound is tight

Extension of Pick's formula to higher dimensions

Abstract

Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$. It follows from the lower bound theorem of Ehrhart polynomials that, when $c > 0$, \[ {\rm vol}(\mathcal{P}) \geq (d \cdot c(\mathcal{P}) + (d-1) \cdot b(\mathcal{P}) - d^2 + 2)/d!, \] where ${\rm vol}(\mathcal{P})$ is the (Lebesgue) volume of $\mathcal{P}$. Pick's formula guarantees that, when $d = 2$, the above inequality is an equality. In the present paper several classes of lattice polytopes for which the equality here holds will be presented.