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A comparison theorem for f-vectors of simplicial polytopes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-7497-2764
2007 (English)In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, Vol. 3, no 1, 347-356 p.Article in journal (Refereed) Published
Abstract [en]

Let f(i)(P) denote the number of i-dimensional faces of a convex polytope P. Furthermore, let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Our main result is that for every simplicial d-polytope P, if f(r) (S (n(1), d)) <= f(r) (P) <= f(r) (C (n(2), d)) for some integers n(1), n(2) and r, then f(s) (S (n(1), d)) <= f(s) (P) <= f(s) (C (n(2), d)) for all s such that r < s. For r = 0 these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain comparison theorem for f-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.

Place, publisher, year, edition, pages
2007. Vol. 3, no 1, 347-356 p.
Keyword [en]
bound conjecture, convex polytope, faces, number
URN: urn:nbn:se:kth:diva-16715ISI: 000247363300012ScopusID: 2-s2.0-49649125635OAI: diva2:334758
QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

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