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 “3d-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.
References
A’Campo-Neuen, A.: On generalized h-vectors of rational polytopes with a symmetry of prime order. Discrete Comput. Geom. 22, 259–268 (1999)
A’Campo-Neuen, A.: On toric h-vectors of centrally symmetric polytopes. Arch. Math. (Basel) 87, 217–226 (2006)
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)
Bayer, M.M.: The extended f-vectors of 4-polytopes. J. Comb. Theory Ser. A 44, 141–151 (1987)
Bayer, M.M., Billera, L.J.: Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets. Invent. Math. 79, 143–157 (1985)
Braden, T.: Remarks on the combinatorial intersection cohomology of fans. Pure Appl. Math. 2, 1149–1186 (2006)
Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973)
Gelfand, I.M., MacPherson, R.D.: Geometry in Grassmannians and a generalization of the dilogarithm. Adv. Math. 44, 279–312 (1982)
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston (1994)
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)
Hanner, O.: Intersections of translates of convex bodies. Math. Scand. 4, 67–89 (1956)
Hansen, A.B.: On a certain class of polytopes associated with independence systems. Math. Scand. 41, 225–241 (1977)
Kalai, G.: Rigidity and the lower bound theorem I. Invent. Math. 88, 125–151 (1987)
Kalai, G.: The number of faces of centrally-symmetric polytopes. Graphs Comb. 5, 389–391 (1989)
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
McMullen, P.: Weights on polytopes. Discrete Comput. Geom. 15, 363–388 (1996)
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)
Novik, I.: The lower bound theorem for centrally symmetric simple polytopes. Mathematika 46, 231–240 (1999)
Paffenholz, A., Ziegler, G.M.: The E t -construction for lattices, spheres and polytopes. Discrete Comput. Geom. 32, 601–624 (2004)
Roth, B.: Rigid and flexible frameworks. Am. Math. Mon. 88, 6–21 (1981)
Sanyal, R.: Constructions and obstructions for extremal polytopes. Ph.D. thesis, TU Berlin (2008)
Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Vol. B. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003). Matroids, trees, stable sets, Chaps. 39–69
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
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)
Stanley, R.: On the number of faces of centrally-symmetric simplicial polytopes. Graphs Comb. 3, 55–66 (1987)
Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. AMS, Providence (1996)
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/
Whiteley, W.: Infinitesimally rigid polyhedra. I. Statics of frameworks. Trans. Am. Math. Soc. 285, 431–465 (1984)
Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)
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Sanyal, R., Werner, A. & Ziegler, G.M. On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes. Discrete Comput Geom 41, 183–198 (2009). https://doi.org/10.1007/s00454-008-9104-8
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DOI: https://doi.org/10.1007/s00454-008-9104-8
Keywords
- Centrally symmetric convex polytopes
- f-vector inequalities
- Flag vectors
- Kalai’s 3d-conjecture
- Equivariant rigidity
- Hanner polytopes
- Hansen polytopes
- Central hypersimplices