The finiteness threshold width of lattice polytopes

Abstract:

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 lattice $d$-polytopes with $n$ lattice points have lattice width at most $w^\infty (d)$. We call $w^\infty (d)$ the finiteness threshold width in dimension $d$ 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. (This last statement was claimed by Barile et al. in [Proc. Am. Math. Soc. 139 (2011), pp. 4247–4253], but we have found a gap in their proof.)

Our main tool is the study of $d$-dimensional lifts of hollow $(d-1)$-polytopes.