Regular Article
Inner Diagonals of Convex Polytopes,☆☆

TO BRANKO GRÜNBAUM IN HONOR OF HIS SEVENTIETH BIRTHDAY
https://doi.org/10.1006/jcta.1998.2953Get rights and content
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Abstract

An inner diagonal of a polytope P is a segment that joins two vertices of P and that lies, except for its ends, in P's relative interior. The paper's main results are as follows: (a) Among all d-polytopes P having a given number v of vertices, the maximum number of inner diagonals is [[formula]]−dv+[[formula]]; when d⩾4 it is attained if and only if P is a stacked polytope. (b) Among all d-polytopes having a given number f of facets, the maximum number of inner diagonals is attained by (and, at least when d=3 and f⩾6, only by) certain simple polytopes. (c) When d=3, the maximum in (b) is determined for all f; when f⩾14 it is 2f2−21f+64 and the unique associated p-vector is 5126f−12. (d) Among all simple 3-polytopes with f facets, the minimum number of inner diagonals is f2−9f+20; when f⩾9 the unique associated p-vector is 324f−4 (f−1)2 and the unique associated combinatorial type is that of the wedge over an (f−1)-gon.

Keywords

convex polytope
3-polytope
d-polytope
simple
simplicial
combinatorial type
p-vector
i-diagonal
inner diagonal
estranged
pulling
pushing
wedge
minimum
maximum

Cited by (0)

Research of the first author was supported by the Natural Science and Engineering Research Council of Canada. Research of the second author was supported in part by the National Science Foundation, U.S.A. We are indebted to Shawn Cokus, Branko Grünbaum, Fred Holt, and Bernhard von Stengel for helpful comments, and to David Avis for supplying software (see [Av]) to enumerate 3-connected planar triangulations.

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