Higher integrality conditions, volumes and Ehrhart polynomials (Q626117)

A polytope is integral if all of its vertices are lattice points. In previous work, the author showed that the coefficients of the Ehrhart polynomial (counting integer points in the \(n\)-tuple polytopes) of a lattice-face polytope are volumes of projections of the polytope. In the current paper the author generalizes both results for certain \(k\)-integral polytopes. It is shown that the Ehrhart polynomial of a \(k\)-integral polytope \(P\) has the properties that the coefficients in degrees less than or equal to \(k\) are determined by a projection of \(P\), and the coefficients in higher degrees are determined by slices of \(P\).