Lattice zonotopes of degree 2

Abstract

AbstractTheEhrhart polynomial$${\text {ehr}}_P (n)$$ehrP(n)of a lattice polytopePgives the number of integer lattice points in then-th dilate ofPfor all integers$$n\ge 0$$n≥0. ThedegreeofPis defined as the degree of its$$h^*$$h∗-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. Azonotopeis the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.