1. Introduction The systematic study of flag f-vectors of polytopes was initiated by Bayer and Billera [1985]. Billera then suggested the study of flag f-vectors of zonotopes; see the dissertation of his student Liu [1995]. The essential computational results of the field appeared in two papers by Billera, Ehrenborg and Readdy [Billera et al. 1997; 1998]. Here we present two classes of linear inequalities for the flag f-vectors of zonotopes. These classes are motivated by our recent results for polytopes [Ehrenborg 2005].

The flag f-vector of a convex polytope contains all the enumerative incidence information between the faces of the polytope. For an n-dimensional polytope the flag f-vector consists of 2n entries; in other words, the flag f-vector lies in the vector space ℝ2 . Bayer and Billera [1985] showed that the flag vectors of n-dimensional polytopes span a subspace of ℝ2n , called the generalized Dehn- Sommerville subspace and denoted by GDSSn. Bayer and Klapper [1991] proved that GDSSn is naturally isomorphic to the n-th homogeneous component of the noncommutative ring ℝ(c, d), where the grading is given by deg c = 1 and deg d = 2. Hence, the flag f-vector of a polytope P can be encoded by a noncommutative polynomial in the variables c and d, called the cd-index.