The minimum period of the Ehrhart quasi-polynomial of a rational polytope

If P ⊂ R is a rational polytope, then iP (n) := #(nP ∩Z ) is a quasipolynomial in n, called the Ehrhart quasi-polynomial of P . The period of iP (n) must divide D(P ) = min{n ∈ Z>0 : nP is an integral polytope}. Few examples are known where the period is not exactly D(P ). We show that for any D, there is a 2-dimensional triangle P such that D(P ) = D but such that the period of iP (n) is 1, that is, iP (n) is a polynomial in n. We also characterize all polygons P such that iP (n) is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.