A Combinatorial Perspective on the Non-Radon Partitions

Let E be a finite set of points in IWd. Then {A, EA} is a non-Radon partition of E i f f there is a hyperplane H separating A strictly from E A. Or equivalently i f f ,-0 is an acyclic reorientation of (M,,(E), 0). the oriented matroid canonically determined by E. If (M(E), (0) is an oriented matroid without loops then the set NR(E, fl) = {(A, E-A): ,-B is acyclic} determines (M(E), 0). In particular the matroidal properties of a finite set of points in IWd are precisely the properties which can be formulated in non-Radon partitions terms. The Mobius function of the poset .01= {A: A G E, ,-I? is acyclic) and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of aB” dissected by hyperplanes, Trans. Amer. Math. Sot. 283 (1984), no. 2, 617631. I(‘, 1985 Academic Press, Inc.