Entering and leaving j-facets

Abstract

Let S be a set of n points in d-space, no i + 1 points on a common (i - 1)-flat for 1 ≤ i ≤ d. An oriented (d - 1)-simplex spanned by d points in S is called a j-facet of S if there are exactly j points from S on the positive side of its affine hull. We show: (*) For j ≤ n/2 - 2, the total number of (≤ j)-facets (i.e. the number of i-facets with 0 ≤ i ≤ j) in 3-space is maximized in convex position (where these numbers are known). A large part of this presentation is a preparatory review of some basic properties of the collection of j-facets - some with their proofs - and of relations to well-established concepts and results from the theory of convex polytopes (h-vector, Dehn-Sommerville relations, Upper Bound Theorem, Generalized Lower Bound Theorem). The relations are established via a duality closely related to the Gale transform - similar to previous works by Lee, by Clarkson, and by Mulmuley. A central definition is as follows. Given a directed line ℓ and a j-facet F of S, we say that ℓ enters F if ℓ intersects the relative interior of F in a single point, and if ℓ is directed from the positive to the negative side of F. One of the results reviewed is a tight upper bound of (j+d-1d-1) on the maximum number of j-facets entered by a directed line. Based on these considerations, we also introduce a vector for a point relative to a point set, which - intuitively speaking - expresses "how interior" the point is relative to the point set. This concept allows us to show that statement (*) above is equivalent to the Generalized Lower Bound Theorem for d-polytopes with at most d + 4 vertices.