Ordering Points by Linear Functionals

Given a set of points in Euclidean space, we say that two linear functionals diier on that set if they give rise to diierent linear orderings of the points. We investigate what the largest and smallest number of diierent linear functionals can be as a function of the number of points and the dimension of the space. 1. Introduction The purpose of this paper is to investigate the number of diierent linear orderings of point conngurations which can arise from linear func-tionals. This problem is related to work of Ungar 5] on the minimum number of directions that a set of points in the plane determine. This investigation has a number of interesting features. First, it seems to be a natural question to ask. Second, it provides a framework in which to extend the work of Ungar 5], which currently is almost a curiosity (although a very beautiful one.) Our results can be applied to give bounds on the number of monotone paths on polytopes. Finally, we think that the technique used to establish the upper bound may have broader applicability and should be more widely known. The structure of the paper is as follows. In the next section we establish our terminology and prove the fundamental connections between the various objects we study. The knowledgeable reader will recognize a number of these constructions from matroid theory, but we have de