A partition theorem for Euclidean $n$-space

Let Vn be an n-dimensional vector space over the reals. Let s . . ' Sins 7)1, * * I, nbe 2n vectors in Vn such that every sequence of vectors { a,, , an }, where ai is either ti or vi, is a linearly independent set. Let (a,, * *, I n) denote the cone spanned by these a's, i.e. the set of all linlear combinations of the ai with non-negative coefficients. The 2n cones in Vn spanlned by the 2n such sequences of a's will be said to partition Vn if their union is all of Vn and if the intersection of every pair of distinct cones (axi, , an), (at; , a*t) is precisely the lower-dimensional cone ("face," "edge," etc.) spanned by the common vectors, i.e. by those ai for which ai = a!, or I 01 if there are no such ai. We may and shall assume that Vn is the usual Euclidean space En of ordered n-tuples of reals. The most familiar example of a partition is the one with the unit vectors (1, 0, 0, , 0), (0, 1, 0, , 0), * ,(0, ** 0, 0, 1) in the role of the i's, and their negatives in the role of the a's. The partitioning cones are then the 2n orthants into which the rectangular coordinate axes divide the space. In this paper, a simple necessary and sufficient condition that the ('s and 77's provide a partition is given; the result is reinterpreted in the language of matrices. Let the orientation of an ordered set of vectors {la,, a2, a * be given as usual by the sign of the determinant, sgn det (ai, , an). Any n -1 of the a's determine a hyperplane, separating En into the two half-spaces lying on opposite sides of the hyperplane. For the sake of simplicity in the statement of the theorem, we assume without loss of generality that the ordered set {n , I2* , ^ } is positively oriented.