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On the generalized lower bound conjecture for polytopes and spheres

Satoshi Murai Department of Mathematical Science, Faculty of Science, Yamaguchi University Eran Nevo Department of Mathematics, Ben Gurion University of the Negev

Combinatorics Arithmetic Geometry and Commutative Algebra mathscidoc:1701.06001

Acta Mathematica, 210, (1), 185-202, 2012.4
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 ($h$_{0},$h$_{1}, …,$h$_{$d$}) satisfies $$ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $$ . Moreover, if$h$_{$r$−1}=$h$_{$r$}for some $$ r\leq \frac{1}{2}d $$ then$P$can be triangulated without introducing simplices of dimension ≤$d$−$r$.
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@inproceedings{satoshi2012on,
  title={On the generalized lower bound conjecture for polytopes and spheres},
  author={Satoshi Murai, and Eran Nevo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400559845769},
  booktitle={Acta Mathematica},
  volume={210},
  number={1},
  pages={185-202},
  year={2012},
}
Satoshi Murai, and Eran Nevo. On the generalized lower bound conjecture for polytopes and spheres. 2012. Vol. 210. In Acta Mathematica. pp.185-202. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400559845769.
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