Combinatorial face enumeration in arrangements and oriented matroids

Abstract

Let f k ( 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: f k ( F )≤( d k ) f d ( 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 P d or in Euclidean space E d . 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.