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2013 On the generalized lower bound conjecture for polytopes and spheres
Satoshi Murai, Eran Nevo
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Acta Math. 210(1): 185-202 (2013). DOI: 10.1007/s11511-013-0093-y

Abstract

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h0, h1, …, hd) satisfies $ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $. Moreover, if hr−1 = hr for some $ r\leq \frac{1}{2}d $ then P can be triangulated without introducing simplices of dimension ≤dr.

The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

Citation

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Satoshi Murai. Eran Nevo. "On the generalized lower bound conjecture for polytopes and spheres." Acta Math. 210 (1) 185 - 202, 2013. https://doi.org/10.1007/s11511-013-0093-y

Information

Received: 5 April 2012; Revised: 13 November 2012; Published: 2013
First available in Project Euclid: 31 January 2017

zbMATH: 1279.52014
MathSciNet: MR3037614
Digital Object Identifier: 10.1007/s11511-013-0093-y

Rights: 2013 © Institut Mittag-Leffler

Vol.210 • No. 1 • 2013
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