The upper bound theorem for polytopes: an easy proof of its asymptotic version

Assume P* is a simple, convex, d-polytope with ν facets, and F* is a simple, convex d'-polytope with ν' facets, where 0 ≤ d' ≤ d-1. If F* is in fact a face of P* we call P*, F* a polytope pair of type d, ν, d', ν'. Define Q* to be P* ∼ F*, the unbounded,...

The upper bound inequality[formula](0 i d/2) is proved for the torich-vector of a rational convexd-dimensional polytope withnvertices. This gives nonlinear inequalities on flag vectors of rational polytopes.

We construct a family of cubical polytypes which shows that the upper bound on the number of facets of a cubical polytope (given a fixed number of vertices) is higher than previously suspected. We also formulate a lower bound conjecture for cubical ...