In each dimension
d
d
there is a constant
w
∞
(
d
)
∈
N
w^\infty (d)\in \mathbb {N}
such that for every
n
∈
N
n\in \mathbb {N}
all but finitely many lattice
d
d
-polytopes with
n
n
lattice points have lattice width at most
w
∞
(
d
)
w^\infty (d)
. We call
w
∞
(
d
)
w^\infty (d)
the finiteness threshold width in dimension
d
d
and show that
d
−
2
≤
w
∞
(
d
)
≤
O
∗
(
d
4
/
3
)
d-2 \le w^\infty (d)\le O^*\left ( d^{4/3}\right )
.
Blanco and Santos determined the value
w
∞
(
3
)
=
1
w^\infty (3)=1
. Here, we establish
w
∞
(
4
)
=
2
w^\infty (4)=2
. This implies, in particular, that there are only finitely many empty
4
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
d
-dimensional lifts of hollow
(
d
−
1
)
(d-1)
-polytopes.