Abstract
Let S be a set of n points in d-space, no i + 1 points on a common (i−l)-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 \(\left( \begin{gathered} j + d - 1 \hfill \\ d - 1 \hfill \\ \end{gathered} \right)\) 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.
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Welzl, E. Entering and leaving j-facets. Discrete Comput Geom 25, 351–364 (2001). https://doi.org/10.1007/s004540010085
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DOI: https://doi.org/10.1007/s004540010085