On the generalized lower bound conjecture for polytopes and spheres

Satoshi Murai, Eran Nevo*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Scopus citations


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, Moreover, if hr-1 = hr for some, 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.

Original languageAmerican English
Pages (from-to)185-202
Number of pages18
JournalActa Mathematica
Issue number1
StatePublished - 2013
Externally publishedYes

Bibliographical note

Funding Information:
Research of the first author was partially supported by KAKENHI 22740018. Research of the second author was partially supported by Marie Curie grant IRG-270923 and by an ISF grant.


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