The Finiteness Threshold Width of Lattice Polytopes

TL;DR

This paper establishes the finiteness threshold width for lattice polytopes in each dimension, showing that beyond a certain width, only finitely many polytopes exist with a given number of lattice points.

Contribution

It proves the existence and bounds of the finiteness threshold width for lattice polytopes in all dimensions, extending previous results in dimension three to four.

Findings

W^ fty(3)=1

W^ fty(4)=2

Finiteness of empty 4-simplices of width greater than two

Abstract

We prove that in each dimension $d$ there is a constant $w^\infty(d)\in \mathbb{N}$ such that for every $n\in \mathbb{N}$ all but finitely many $d$-polytopes with $n$ lattice points have width at most $w^\infty(d)$. We call $w^\infty(d)$ the finiteness threshold width and show that $d-2 \le w^\infty(d)\le O^*\left( d^{4/3}\right)$. Blanco and Santos determined the value $w^\infty(3)=1$. Here, we establish $w^\infty(4)=2$. This implies, in particular, that there are only finitely many empty $4$-simplices of width larger than two. The main tool in our proofs is the study of $d$-dimensional lifts of hollow $(d-1)$-polytopes.