The Finiteness Threshold Width of Lattice Polytopes

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 $d-2\le w^{\infty }(d)\le O^{\ast }\left(d^{4/3}\right)$. 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.

21 pages, 10 figures; changes from previous version: several edits suggested by anonymous referees; this version has been accepted in Trans. of the Amer. Math. Soc

References in corpus (4)

• Lattice 3-polytopes with six lattice points
• Enumeration of lattice 3-polytopes by their number of lattice points
• The complete classification of empty lattice $4$-simplices
• Classification of empty lattice $4$-simplices of width larger than two

Cited by in corpus (1)

• Blowups with log canonical singularities