A complete h-vector for convex polytopes

This note defines a complete h-vector for convex polytopes, which extends the already known toric (or mpih) h-vector and has many similar properties. Complete means that it encodes the whole of the flag vector. First we define the concept of a generalised h-vector and state some properties that follow. The toric h-vector is given as an example. We then define a complete generalised h-vector, and again state properties. Finally, we show that this complete h-vector and all with similar properties will sometimes have negative coefficients. Most of the proofs, and further investigations, will appear elsewhere.Comment: 4 pages, LaTeX, no figure

Axioms for the g-vector of general convex polytopes

McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the extended g-vector. This calculation is done by computer for d=5 and the results analysed. The conjectures imply new linear inequalities on convex polytope flag vectors. Underlying the axioms is a hypothesised higher-order homology extension to middle perversity intersection homology (order-zero homology), which measures the failure of lower-order homology to have a ring structure.Comment: LaTeX2e. 10 page

FLAG \u3cem\u3eF\u3c/em\u3e-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