THE NUMBER OF k-FACES OF A SIMPLE d-POLYTOPE

ANDERS BJÖRNER SVANTE LINUSSON

Abstract. Consider the question: Given integers k < d < n, does there exist a simple d-polytope with n faces of dimension k? We show that there exist numbers G(d, k) and N(d, k) such that for n > N(d, k) the answer is yes if and only if n ≡ 0 (mod G(d, k)). Furthermore, a formula for G(d, k) is given, showing that e.g. G(d, k) = 1 if k ≥ ⌊\frac{d+1}{2}⌋ or if both d and k are even, and also in some other cases (meaning that all numbers beyond N(d, k) occur as the number of k-faces of some simple d-polytope).

This question has previously been studied only for the case of vertices (k = 0), where Lee proved the existence of N(d, 0) (with G(d, 0) = 1 or 2 depending on whether d is even or odd), and Prabhu showed that N(d, 0) ≤ cd\sqrt{d}. We show here that asymptotically the true value of Prabhu’s constant is c = \sqrt{2} if d is even, and c = 1 if d is odd.