Lattice zonotopes of degree 2

Abstract

The Ehrhart polynomial$${\text {ehr}}_P (n)$$ ehr P ( n ) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers $$n\ge 0$$ n ≥ 0 . The degree of P is 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. A zonotope is 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.