The Finiteness Threshold Width of Lattice Polytopes

We prove that in each dimension $d$ there is a constant $w^\infty(d)\in $\mathbb{N}$$suchthatforevery$n\in $\mathbb{N}$$allbutfinitelymany$d$-polytopeswith$n$latticepointshavewidthatmost$w^\infty(d)$\mathrm{.}Wecall$w^\infty(d)$thefinitenessthresholdwidthandshowthat$d-2 \le w^\infty(d)\le O^*\left( d^{4/3}\right)$. Blanco and Santos determined the value $w^{\mathrm{\infty }}(3)=1$. Here, we establish $w^{\mathrm{\infty }}(4)=2$. 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.