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2004 On the Monotone Upper Bound Problem
Julian Pfeifle, Günter M. Ziegler
Experiment. Math. 13(1): 1-12 (2004).

Abstract

The Monotone Upper Bound Problem asks for the maximal number {\small $M(d,n)$} of vertices on a strictly increasing edge-path on a simple {\small $d$}-polytope with {\small $n$} facets. More specifically, it asks whether the upper bound

{\small \[ M(d,n)\ \le\ M_{\rm ubt}(d,n) \]}\!\!

provided by McMullen's [McMullen 70] Upper Bound Theorem is tight, where {\small $M_{\rm ubt}(d,n)$} is the number of vertices of a dual-to-cyclic d-polytope with n facets.

It was recently shown that the upper bound {\small $M(d,n)\le M_{\rm ubt}(d,n)$} holds with equality for small dimensions ({\small $d\le 4$} [Pfeifle 04]) and for small corank ({\small $n\le d+2$} [Gärtner et al. 01]). Here we prove that it is not tight in general: in dimension {\small $d=6$}, a polytope with {\small $n=9$} facets can have {\small $ M_{\rm ubt}(6,9)=30$} vertices, but not more than {\small $M(6,9)\le29$} vertices can lie on a strictly increasing edge-path.

The proof involves classification results about neighborly polytopes of small corank, Kalai's [Kalai 88] concept of abstract objective functions, the Holt-Klee conditions [Holt and Klee 98], explicit enumeration, Welzl's extended Gale diagrams [Welzl 01], and randomized generation of instances, as well as nonrealizability proofs via a version of the Farkas lemma.

Citation

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Julian Pfeifle. Günter M. Ziegler. "On the Monotone Upper Bound Problem." Experiment. Math. 13 (1) 1 - 12, 2004.

Information

Published: 2004
First available in Project Euclid: 10 June 2004

zbMATH: 1068.52019
MathSciNet: MR2065564

Subjects:
Primary: 52B55
Secondary: 05C20‎, 05C30, 52B35

Rights: Copyright © 2004 A K Peters, Ltd.

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Vol.13 • No. 1 • 2004
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