  The Ehrhart polynomial of a lattice polytope $P$ encodes information about
the number of integer lattice points in positive integral dilates of $P$. The
$h^\ast$-polynomial of $P$ is the numerator polynomial of the generating
function of its Ehrhart polynomial. A zonotope is any projection of a higher
dimensional cube. We give a combinatorial description of the
$h^\ast$-polynomial of a lattice zonotope in terms of refined descent
statistics of permutations and prove that the $h^\ast$-polynomial of every
lattice zonotope has only real roots and therefore unimodal coefficients.
Furthermore, we present a closed formula for the $h^\ast$-polynomial of a
zonotope in matroidal terms which is analogous to a result by Stanley (1991) on
the Ehrhart polynomial. Our results hold not only for $h^\ast$-polynomials but
carry over to general combinatorial positive valuations. Moreover, we give a
complete description of the convex hull of all $h^\ast$-polynomials of
zonotopes in a given dimension: it is a simplicial cone spanned by refined
Eulerian polynomials.

Combinatorics;Metric Geometry