Advertisement

Discrete & Computational Geometry

, Volume 41, Issue 2, pp 183–198 | Cite as

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

  • Raman Sanyal
  • Axel Werner
  • Günter M. Ziegler
Article

Abstract

In 1989, Kalai stated three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the “3 d -conjecture.” It is well known that the three conjectures hold in dimensions d≤3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d≥5.

Keywords

Centrally symmetric convex polytopes f-vector inequalities Flag vectors Kalai’s 3d-conjecture Equivariant rigidity Hanner polytopes Hansen polytopes Central hypersimplices 

References

  1. 1.
    A’Campo-Neuen, A.: On generalized h-vectors of rational polytopes with a symmetry of prime order. Discrete Comput. Geom. 22, 259–268 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A’Campo-Neuen, A.: On toric h-vectors of centrally symmetric polytopes. Arch. Math. (Basel) 87, 217–226 (2006) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bárány, I., Lovász, L.: Borsuk’s theorem and the number of facets of centrally symmetric polytopes. Acta Math. Acad. Sci. Hung. 40, 323–329 (1982) zbMATHCrossRefGoogle Scholar
  4. 4.
    Bayer, M.M.: The extended f-vectors of 4-polytopes. J. Comb. Theory Ser. A 44, 141–151 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bayer, M.M., Billera, L.J.: Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets. Invent. Math. 79, 143–157 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Braden, T.: Remarks on the combinatorial intersection cohomology of fans. Pure Appl. Math. 2, 1149–1186 (2006) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973) Google Scholar
  8. 8.
    Gelfand, I.M., MacPherson, R.D.: Geometry in Grassmannians and a generalization of the dilogarithm. Adv. Math. 44, 279–312 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston (1994) zbMATHGoogle Scholar
  10. 10.
    Grünbaum, B.: Convex Polytopes, 2nd edn. Graduate Texts in Mathematics, vol. 221. Springer, New York (2003). Second edition by V. Kaibel, V. Klee and G.M. Ziegler (original edition: Interscience, London 1967) Google Scholar
  11. 11.
    Hanner, O.: Intersections of translates of convex bodies. Math. Scand. 4, 67–89 (1956) MathSciNetGoogle Scholar
  12. 12.
    Hansen, A.B.: On a certain class of polytopes associated with independence systems. Math. Scand. 41, 225–241 (1977) MathSciNetGoogle Scholar
  13. 13.
    Kalai, G.: Rigidity and the lower bound theorem I. Invent. Math. 88, 125–151 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kalai, G.: The number of faces of centrally-symmetric polytopes. Graphs Comb. 5, 389–391 (1989) CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Kuperberg, G.: From the Mahler conjecture to Gauss linking integrals. Preprint, Oct. 2006, 9 p., version 3, 10 p., July 2008, http://arxiv.org/abs/math/0610904v3
  16. 16.
    McMullen, P.: Weights on polytopes. Discrete Comput. Geom. 15, 363–388 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Moon, J.W.: Some enumerative results on series-parallel networks. In: Random Graphs ’85, Poznań, 1985. North-Holland Math. Stud., vol. 144, pp. 199–226. North-Holland, Amsterdam (1987) Google Scholar
  18. 18.
    Novik, I.: The lower bound theorem for centrally symmetric simple polytopes. Mathematika 46, 231–240 (1999) zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Paffenholz, A., Ziegler, G.M.: The E t-construction for lattices, spheres and polytopes. Discrete Comput. Geom. 32, 601–624 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Roth, B.: Rigid and flexible frameworks. Am. Math. Mon. 88, 6–21 (1981) zbMATHCrossRefGoogle Scholar
  21. 21.
    Sanyal, R.: Constructions and obstructions for extremal polytopes. Ph.D. thesis, TU Berlin (2008) Google Scholar
  22. 22.
    Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Vol. B. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003). Matroids, trees, stable sets, Chaps. 39–69 Google Scholar
  23. 23.
    Sloane, N.J.A.: Number of series-parallel networks with n unlabeled edges, multiple edges not allowed. Sequence A058387, The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/A058387
  24. 24.
    Stanley, R.: Generalized H-vectors, intersection cohomology of toric varieties, and related results. In: Commutative Algebra and Combinatorics, Kyoto, 1985. Adv. Stud. Pure Math., vol. 11, pp. 187–213. North-Holland, Amsterdam (1987) Google Scholar
  25. 25.
    Stanley, R.: On the number of faces of centrally-symmetric simplicial polytopes. Graphs Comb. 3, 55–66 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. AMS, Providence (1996) zbMATHGoogle Scholar
  27. 27.
    Tao, T.: Open question: the Mahler conjecture on convex bodies. Blog page started 8 March, 2007, http://terrytao.wordpress.com/2007/03/08/open-problem-the-mahler-conjecture-on-convex-bodies/
  28. 28.
    Whiteley, W.: Infinitesimally rigid polyhedra. I. Statics of frameworks. Trans. Am. Math. Soc. 285, 431–465 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Raman Sanyal
    • 1
  • Axel Werner
    • 1
  • Günter M. Ziegler
    • 1
  1. 1.Institute of Mathematics, MA 6-2TU BerlinBerlinGermany

Personalised recommendations