Combinatorial face enumeration in arrangements and oriented matroids

Abstract

Let fk(F) denote the number of k-dimensional faces of a d-dimensional arrangement F of spheres or a d-dimensional oriented matroid F. In this paper we show that the following relation among the face numbers is valid: fk(F)≤(dk)fd(F) for 0≤k≤d. The same inequalities are valid for d-dimensional arrangements of hyperplanes. Using the result, we obtain a polynomial algorithm to enumerate all faces from the set of maximal faces of an oriented matroid. This algorithm can be applied to any arrangement of hyperplanes in projective space Pd or in Euclidean space Ed. Combining this with a recent result of Cordovil and Fukuda, we have the following: given the cograph of an arrangement (where the vertices are the d-faces and two vertices are adjacent if they intersect in a (d-1)-face), one can reconstruct the location vectors of all faces of the arrangement up to isomorphism in polynomial time. It is also shown that one can test in polynomial time whether a given set of (+,0,-)-vectors is the set of maximal vectors (topes) of an oriented matroid. © 1991.