Ehrhart Quasi-Polynomials of almost integral polytopes

In this thesis we characterize centrally symmetric lattice polytopes and lattice zonotopes through properties of the Ehrhart quasi-polynomials of almost integral polytopes. To this end, we introduce the notion of GCD-property and symmetry for quasi-polynomials. A lattice polytope is centrally symmetric if and only if the Ehrhart quasi-polynomial of every almost integral polytope derived from that polytope is symmetric. Furthermore, we show that a lattice polytope is a zonotope if and only if the Ehrhart quasi-polynomial of every almost integral polytope derived from that polytope satisfies the GCD-property. In order to describe the constituents of the Ehrhart quasi-polynomial of an almost integral polytope, we introduce the translated lattice point enumerator and prove that this function is a polynomial.