Two combinatorial properties of a class of simplicial polytopes
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Letf(Psd) be the set of allf-vectors of simpliciald-polytopes. ForP a simplicial 2d-polytope let Σ(P) denote the boundary complex ofP. We show that for eachf ∈f(Psd) there is a simpliciald-polytopeP withf(P)=f such that the 11 02 simplicial diameter of Σ(P) is no more thanf0(P)−d+1 (one greater than the conjectured Hirsch bound) and thatP admits a subdivision into a simpliciald-ball with no new vertices that satisfies the Hirsch property. Further, we demonstrate that the number of bistellar operations required to obtain Σ(P) from the boundary of ad-simplex is minimum over the class of all simplicial polytopes with the samef-vector. This polytopeP will be the one constructed to prove the sufficiency of McMullen's conditions forf-vectors of simplicial polytopes.
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- 6.C. W. Lee,Counting the faces of simplicial convex polytopes, Ph.D. Thesis, Cornell University, Ithaca, New York, 1981.Google Scholar
- 7.P. McMullen and G. C. Shephard,Convex Polytopes and the Upper Bound Conjecture, London Math. Soc. Lecture Note Series 3, Cambridge University Press, 1971.Google Scholar
- 10.R. P. Stanley,Cohen-Macaulay complexes, inHigher Combinatorics (M. Aigner, ed.), D. Reidel, Dordrecht-Holland, 1977, pp. 51–62.Google Scholar