An Upper Bound Theorem for Polytope Pairs

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, simple d-polyhedron obtained by applying a protective transformation that sends a supporting hyperplane for F* onto the hyperplane at infinity. In this paper we answer the question: What are the maximum possible numbers of faces of different dimensions that P* and Q* can have? We restate and solve the problem in a dual, simplicial context.