On the generalized lower bound conjecture for polytopes and spheres

Murai, Satoshi; Nevo, Eran
March 2013
Acta Mathematica;Mar2013, Vol. 210 Issue 1, p185
Academic Journal
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 ( h, h, ..., h) satisfies $$ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $$. Moreover, if h = h 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.


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