## Acta Mathematica

### On the generalized lower bound conjecture for polytopes and spheres

#### 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.

#### Article information

Source
Acta Math., Volume 210, Number 1 (2013), 185-202.

Dates
Revised: 13 November 2012
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892675

Digital Object Identifier
doi:10.1007/s11511-013-0093-y

Mathematical Reviews number (MathSciNet)
MR3037614

Zentralblatt MATH identifier
1279.52014

Rights

#### Citation

Murai, Satoshi; Nevo, Eran. On the generalized lower bound conjecture for polytopes and spheres. Acta Math. 210 (2013), no. 1, 185--202. doi:10.1007/s11511-013-0093-y. https://projecteuclid.org/euclid.acta/1485892675

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