On the generalized lower bound conjecture for polytopes and spheres

By Satoshi Murai, Eran Nevo. 2012

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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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d342d2afc7bb32f33c158df8901be8f818d54930