Lattice 3-polytopes with five lattice points

Mónica Blanco, Francisco Santos

Published 2014-09-23Version 1

We work out the complete classification of lattice $3$-polytopes with exactly $5$ lattice points. We first show that for each $n$ there is only a finite number of (equivalence classes of) $3$-polytopes of lattice width larger than one. Since polytopes of width one are easy to classify, we concentrate on an exhaustive classification of those of width larger than one. For $n=4$, all empty tetrahedra have width one (White). For $n=5$ we show that there are exactly 9 different polytopes of width $2$, and none of larger width. Eight of them are the clean tetrahedra previously classified by Kasprzyk and (independently) Reznick.