FLAG F-VECTORS OF POLYTOPES WITH FEW VERTICES

Abstract

OF DISSERTATION FLAG F -VECTORS OF POLYTOPES WITH FEW VERTICES We may describe a polytope P as the convex hull of n points in space. Here we consider the numbers of chains of faces of P . The toric g-vector and CD-index of P are useful invariants for encoding this information. For a simplicial polytope P , Lee defined the winding number wk in a Gale diagram corresponding to P . He showed that wk in the Gale diagram equals gk of the corresponding polytope. In this dissertation, we fully establish how to compute the g-vector for any polytope with few vertices from its Gale diagram. Further, we extend these results to polytopes with higher dimensional Gale diagrams in certain cases, including the case when all the points are in affinely general position. In the Generalized Lower Bound Conjecture, McMullen and Walkup predicted that if gk(P ) = 0 for some simplicial polytope P and some k, then P is (k− 1)-stacked. Lee and Welzl independently use Gale transforms to prove the GLBC holds for any simplicial polytope with few vertices. In the context of Gale transforms, we will extend this result to nonpyramids with few vertices. We will also prove how to obtain the CD-index of polytopes dual to polytopes with few vertices in several cases. For instance, we show how to compute the CD-index of a polytope from the Gale diagram of its dual polytope when the Gale diagram is 2-dimensional and the origin is captured by a line segment.