On lattice points in polyhedral cross-sections (Q2368133)

The authors prove that the convex hull of any \(k^ d+1\) points of a \(d\)- dimensional lattice contains \(k+1\) collinear lattice points. The authors further consider for a convex polytope the number of its lattice points in consecutive parallel lattice hyperplanes (levels). They prove the surprising result that, if the polytope spans at most \(2^{d- 1}\) levels, then its string of numbers may be arbitrary. In contrast to this they give a string of \(2^{d-1}+1\) of numbers which cannot be associated to a convex polytope in \(\mathbb{R}^ d\).