| Here we study the combinatorics of polytopes. A polytope P is the convex hull of a finite set of points in R d, and its boundary is a collection of lower-dimensional polytopes known as the faces of P. The flag f-vector of P counts the faces of each dimension and their incidences with one another. We would like to know what linear inequalities the entries of the flag f-vector satisfy.; First we present some of the history of this problem, along with the necessary mathematical background. We discuss several special classes of polytopes, including simplicials, simples, cubicals, and zonotopes, whose flag f-vectors satisfy inequalities not satisfied by all polytopes.; Then we define Stanley's toric g-vector, which can be used to generate many linear inequalities for flag f-vectors. We prove Meisinger's conjecture that some of these inequalities are implied by others. In addition, we consider the cd-index, another source of many inequalities. We show that not all of these are consequences of the non-negativity of the toric g-vector.; We then use linear inequalities satisfied by lower-dimensional polytopes to generate linear relations satisfied by simplicial, simple, k-simplicial, and k-simple polytopes and cubical zonotopes. We also examine a g-vector for cubical polytopes proposed by Adin and give evidence that supports the conjecture g2 ≥ 0. In particular, we show this to hold for the class of almost simple cubical polytopes, where one might expect it is most likely to fail. Next we improve upon previously known linear inequalities satisfied by zonotopes. Finally, we construct examples of another special class of polytopes, the self-dual polytopes. |
