The finiteness threshold width of lattice polytopes

Abstract

In each dimension d there is a constant woo(d) [épsilon] N such that for every n [épsilon] N all but finitely many lattice d-polytopes with n lattice points have llattice width at most woo(d). We call woo(d) the finiteness threshold width in dimension d and show that d - 2 woo(d) O _d4/3_. Blanco and Santos determined the value woo(3) = 1. Here, we establish woo(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.