Ehrhart polynomials of cyclic polytopes as averages of zonotope Ehrhart polynomials

Source abstract

We prove an averaging formula for the Ehrhart polynomial of a cyclic polytope whose vertices are given by integer parameters on the moment curve. More precisely, its Ehrhart polynomial is the average of the Ehrhart polynomials of an explicitly constructed family of lattice zonotopes. Since lattice zonotopes are magic positive, this formula implies magic positivity for these cyclic polytopes. Consequently, their $h^\ast$-polynomials are real-rooted, and their $h^\ast$-vectors are log-concave and unimodal.