Acta Mathematica

On the generalized lower bound conjecture for polytopes and spheres

Satoshi Murai and Eran Nevo

Full-text: Open access


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 ≤dr.

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.

Article information

Acta Math., Volume 210, Number 1 (2013), 185-202.

Received: 5 April 2012
Revised: 13 November 2012
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2013 © Institut Mittag-Leffler


Murai, Satoshi; Nevo, Eran. On the generalized lower bound conjecture for polytopes and spheres. Acta Math. 210 (2013), no. 1, 185--202. doi:10.1007/s11511-013-0093-y.

Export citation


  • B agchi, B. & D atta, B., On stellated spheres, shellable balls, lower bounds and a combinatorial criterion for tightness. Preprint, 2011.
  • B arnette, D.W., The minimum number of vertices of a simple polytope. Israel J. Math., 10 (1971), 121–125.
  • — A proof of the lower bound conjecture for convex polytopes. Pacific J. Math., 46 (1973), 349–354.
  • B illera, L. J. & L ee, C. W., A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex polytopes. J. Combin. Theory Ser. A, 31 (1981), 237–255.
  • B runs, W. & H erzog, J., CohenMacaulay Rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993.
  • D ancis, J., Triangulated n-manifolds are determined by their $ \left( {\left\lfloor {{n \left/ {2} \right.}} \right\rfloor +1} \right) $-skeletons. Topology Appl., 18 (1984), 17–26.
  • F lores, A., Über n-dimensionale komplexe die im $ {{\mathbb{R}}^{2n+1 }} $ absolut selbstverschlungen sind. Ergeb. Math. Kolloq. Wien, 6 (1935), 4–6.
  • F ulton, W., Introduction to Toric Varieties. Annals of Mathematics Studies, 131. Princeton University Press, Princeton, NJ, 1993.
  • G oodman, J. E. & P ollack, R., Upper bounds for configurations and polytopes in Rd. Discrete Comput. Geom., 1 (1986), 219–227.
  • G reen, M. L., Generic initial ideals, in Six Lectures on Commutative Algebra (Bellaterra, 1996), Progr. Math., 166, pp. 119–186. Birkhäuser, Basel, 1998.
  • G rünbaum, B., Convex Polytopes. Graduate Texts in Mathematics, 221. Springer, New York, 2003.
  • H erzog, J. & H ibi, T., Monomial Ideals. Graduate Texts in Mathematics, 260. Springer, London, 2011.
  • K alai, G., Rigidity and the lower bound theorem. I. Invent. Math., 88 (1987), 125–151.
  • — Many triangulated spheres. Discrete Comput. Geom., 3 (1988), 1–14.
  • — Some aspects of the combinatorial theory of convex polytopes, in Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 440, pp. 205–229. Kluwer, Dordrecht, 1994.
  • van K ampen, E. R., Komplexe in euklidischen Räumen. Abh. Math. Sem. Hamburg, 9 (1932), 72–78.
  • K leinschmidt, P. & L ee, C.W., On k-stacked polytopes. Discrete Math., 48 (1984), 125–127.
  • M cMullen, P., The numbers of faces of simplicial polytopes. Israel J. Math., 9 (1971), 559–570.
  • — Triangulations of simplicial polytopes. Beitr. Algebra Geom., 45 (2004), 37–46.
  • M cMullen, P. & W alkup, D.W., A generalized lower-bound conjecture for simplicial polytopes. Mathematika, 18 (1971), 264–273.
  • M iller, E. & S turmfels, B., Combinatorial Commutative Algebra. Graduate Texts in Mathematics, 227. Springer, New York, 2005.
  • M unkres, J. R., Elements of Algebraic Topology. Addison–Wesley, Menlo Park, CA, 1984.
  • N agel, U., Empty simplices of polytopes and graded Betti numbers. Discrete Comput. Geom., 39 (2008), 389–410.
  • R eisner, G. A., Cohen–Macaulay quotients of polynomial rings. Adv. Math., 21 (1976), 30–49.
  • R udin, M. E., An unshellable triangulation of a tetrahedron. Bull. Amer. Math. Soc., 64 (1958), 90–91.
  • S hapiro, A., Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction. Ann. of Math., 66 (1957), 256–269.
  • S tanley, R. P., Cohen–Macaulay complexes, in Higher Combinatorics (Berlin, 1976), NATO Adv. Study Inst. Ser. Ser. C Math. Phys. Sci., 31, pp. 51–62. Reidel, Dordrecht, 1977.
  • — The number of faces of a simplicial convex polytope. Adv. Math., 35 (1980), 236–238.
  • S wartz, E., Face enumeration—from spheres to manifolds. J. Eur. Math. Soc. (JEMS), 11 (2009), 449–485.
  • W u, W., A Theory of Imbedding, Immersion, and Isotopy of Polytopes in a Euclidean Space. Science Press, Beijing, 1965.
  • Z iegler, G. M., Lectures on Polytopes. Graduate Texts in Mathematics, 152. Springer, New York, 1995.