A lower bound theorem for polytope pairs
Let (P,F) be a pair of polytopes of class (d,n,u), i.e. P is a simple d- polytope in \({\mathbb{R}}^ d\) with n facets, F is a facet of P such that precisely u facets of P intersect F. If k denote the number of vertices of P not in F, the author gives a sharp upper bound for the number of vertices of the facet F in terms of k, n, and u, thus generalizing a result of \textit{V. Klee} [Acta. Math. 133, 1-25 (1974; Zbl 0307.90042)] who gave a proof for \(u=n-1\). This theorem implies Barnette's result on the lower bound of the number of vertices of a simple d-polytope in terms of the number of its facets.
scientific article; zbMATH DE number 3964868