The minimal volume of a lattice polytope

TL;DR

This paper provides an elementary proof for the minimal volume of a lattice polytope based on its boundary and interior lattice points, refining understanding of lattice polytope volume bounds.

Contribution

It offers a simple, triangulation-based proof of the minimal volume formula for lattice polytopes, improving clarity and accessibility.

Findings

Derived the minimal volume formula for lattice polytopes with interior points.

Established a triangulation method for proving volume bounds.

Clarified the relationship between boundary, interior points, and volume.

Abstract

Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c$ that to the interior of $\mathcal{P}$. It follows from a lower bound theorem of Ehrhart polynomials that, when $c > 0$, the volume of $\mathcal{P}$ is bigger than or equal to $(dc + (d-1)b - d^2 + 2)/d!$. In the present paper, via triangulations, a short and elementary proof of the minimal volume formula is given.