Lattice 3-polytopes with few lattice points

TL;DR

This paper classifies all lattice 3-polytopes with five points, revealing a finite set of classes for width greater than one and extending White's earlier work on empty tetrahedra.

Contribution

It provides a complete classification of lattice 3-polytopes with five points and proves finiteness results for larger sizes and widths.

Findings

Exactly nine classes of 3-polytopes with five points and width two.

Infinite classes of width one polytopes.

Finiteness of classes for any fixed number of points and width > 1.

Abstract

We extend White's classification of empty tetrahedra to the complete classification of lattice $3$-polytopes with five lattice points, showing that, apart from infinitely many of width one, there are exactly nine equivalence classes of them with width two and none of larger width. We also prove that, for each $n\in \mathbb{N}$, there is only a finite number of (classes of) lattice $3$-polytopes with $n$ lattice points and of width larger than one. This implies that extending the present classification to larger sizes makes sense, which is the topic of subsequent papers of ours.