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.