This is something you might be aware of already (and which does not seem to be exactly the situation you are formulating), but Reeve's tetrahedron is a 3-dimensional integral convex polytope without lattice-points in the interior. Furthermore, the only lattice points are the vertices, so there are four of them.
EDIT: I happened to attend a talk by Benjamin Nill a few weeks ago, and part of his talk seemed related to this kind of question. Here is a paper he wrote with Ziegler: http://arxiv.org/abs/arXiv:1101.4292.
