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 ≤d − r.
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
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
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