  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.
