A finite calculus approach to Ehrhart polynomials

A rational polytope is the convex hull of a finite set of points in Rd with rational coordinates. Given a rational polytope P ⊆ Rd, Ehrhart proved that, for t ∈ Z≥0, the function #(tP ∩ Zd) agrees with a quasi-polynomial LP(t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart’s theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen’s theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart–Macdonald theorem on reciprocity. 1 Introduction. Let us first look at an (easy) example of computing a volume. Let ∆d ⊆ R d be the convex hull of the following d + 1 points: the origin and the standard basis vectors ei, 1 ≤ i ≤ d. Let t∆d be its dilation by a factor of t (for nonnegative t). A straightforwar