  An $(n-1)$-tuple $a = (a(1), \dots, a(n-1))$ consisting of positive integers
is said to be asymptotically hollow if there exist infinitely many positive
integers $N$ such that the convex hull, $K(a(n))$, in $n$-dimensional Euclidean
space of $\{ 0,e_1, \dots, e_{n-1}, \alpha(N)^T\}$ is hollow (has no lattice
points in its interior), where $e_i$ run over all but the last standard basis
elements, and $\alpha(N) $ is the row $(a(1), \dots, a(N-1), N)$. The tuple is
trivial if $\min a(i) = 1$. Nontrivial asymptotically hollow tuples are
characterized in terms of modular inequalities, and turn out to be rare. We
show that for a tuple $a$, there exists an effectively computable constant $C$
(depending on $a$) such that if for some $N > C$, $K(\alpha(N))$ is (not)
hollow, then for all $M > C$, $K(\alpha(M))$ is (not) hollow (respectively).
When $n = 4$, the nontrivial asymptotically hollow triples are completely
determined; there are eleven of them, together with a one-(positive
integer)-parameter family.
