A theorem on the average number of subfaces in arrangements and oriented matroids

It is known that for simple arrangements in the d-dimensional Euclidean space RdThe average number of j-dimensional subfaces of a k-dimensional face is less than {Mathematical expression}. In this paper, we show that this is also true for all arrangements in Rd and for all oriented matroids, and we give combinatorial proofs. © 1993 Kluwer Academic Publishers.