The number of faces of simplicial polytopes

Abstract

The numbers of k-dimensional faces, f k≡f k(d), k=−1,0,…,d−1 , of a d-dimensional convex polytope satisfy the relations ∑ i − k d − 1 ( − 1 ) i f i ( i + 1 k + 1 ) = ( − 1 ) d − 1 f k with f −1=1, by convention. These relations are not independent and serve to determine (roughly) half of the f's in terms of the other half. Relations for the f's of upper index in terms of those of lower index found by Branko Grünbaum (in an unpublished report) are rederived here, and recurrences and interrelations of the coefficients in these relations are developed. In addition, the relations for the f's of even index in terms of those of odd index, and vice versa, are found