  We show that: (1) unimodular simplices in a lattice 3-polytope cover a
neighborhood of the boundary of the polytope if and only if the polytope is
very ample, (2) the convex hull of lattice points in every ellipsoid in R^3 has
a unimodular cover, and (3) for every d at least 5, there are ellipsoids in
R^d, such that the convex hulls of the lattice points in these ellipsoids are
not even normal. Part (3) answers a question of Bruns, Michalek, and the
author.
