Inner Diagonals of Convex Polytopes

✍ Scribed by David Bremner; Victor Klee

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 249 KB
• Volume: 87
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1998.2953

✦ Synopsis

to branko gru nbaum in honor of his seventieth birthday 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 ( v 2 )&dv+( d+1 2 ); 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 2f 2 &21f +64 and the unique associated p-vector is 5 12 6 f &12 . (d) Among all simple 3-polytopes with f facets, the minimum number of inner diagonals is f 2 &9f +20; when f 9 the unique associated p-vector is 3 2 4 f &4 ( f &1) 2 and the unique associated combinatorial type is that of the wedge over an ( f &1)-gon.