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

If P ⊂ R d is a rational polytope, then i P ( n ) ≔ # ( nP ∩ Z d ) is a quasi-polynomial in n , called the Ehrhart quasi-polynomial of P . The minimum period of i P ( n ) must divide D ( P ) = min { n ∈ Z > 0 : nP is an integral polytope } . Few examples are known where the minimum 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 minimum period of i P ( n ) is 1, that is, i P ( n ) is a polynomial in n . We also characterize all polygons P such that i P ( 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.