# zbMATH — the first resource for mathematics

On Kalai’s conjectures concerning centrally symmetric polytopes. (English) Zbl 1168.52013
A convex polytope $$P$$ is centrally symmetric if $$P = -P$$. Let $$f_j$$ be the number of $$j$$-dimensional faces of $$P$$. G. Kalai [Graphs Comb. 5, No. 1, 389–391 (1989; Zbl 1168.52303)] posed three conjectures on centrally-symmetric $$d$$-dimensional polytopes, of increasing strength:
Conjecture A states that every centrally-symmetric $$d$$-dimensional polytopes has at least $$3^d$$ nonempty faces, i.e., $$f_0 + f_1 + \cdots + f_{ d-1 } + 1 \geq 3^d$$.
Conjecture B suggests that, given a centrally-symmetric $$d$$-dimensional polytope $$P$$, the face numbers of $$P$$ are bounded below by the face numbers of a Hanner polytope; Hanner polytopes are built recursively from line segments by taking direct sums or direct products.
Conjecture C strengthens Conjecture B by replacing the bounds on the $$f_j$$ by an inequalities involving flag vectors.
It is not hard to see that Kalai’s conjectures hold for $$d \leq 3$$. The authors prove that Conjectures A and B for $$d=4$$, that Conjecture $$B$$ fails for $$d \geq 5$$, and that Conjecture C fails for $$d \geq 4$$.

##### MSC:
 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
OEIS
Full Text:
##### References:
 [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) · Zbl 0940.52006 [2] A’Campo-Neuen, A., On toric $$h$$-vectors of centrally symmetric polytopes, Arch. Math. (Basel), 87, 217-226, (2006) · Zbl 1107.52008 [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) · Zbl 0514.52003 [4] Bayer, M. M., The extended $$f$$-vectors of 4-polytopes, J. Comb. Theory Ser. A, 44, 141-151, (1987) · Zbl 0618.52005 [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) · Zbl 0543.52007 [6] Braden, T., Remarks on the combinatorial intersection cohomology of fans, Pure Appl. Math., 2, 1149-1186, (2006) · Zbl 1110.14019 [7] Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973) · Zbl 0031.06502 [8] Gelfand, I. M.; MacPherson, R. D., Geometry in Grassmannians and a generalization of the dilogarithm, Adv. Math., 44, 279-312, (1982) · Zbl 0504.57021 [9] Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston (1994) · Zbl 0827.14036 [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) [11] Hanner, O., Intersections of translates of convex bodies, Math. Scand., 4, 67-89, (1956) [12] Hansen, A. B., On a certain class of polytopes associated with independence systems, Math. Scand., 41, 225-241, (1977) · Zbl 0411.05033 [13] Kalai, G., Rigidity and the lower bound theorem I, Invent. Math., 88, 125-151, (1987) · Zbl 0624.52004 [14] Kalai, G., The number of faces of centrally-symmetric polytopes, Graphs Comb., 5, 389-391, (1989) · Zbl 1168.52303 [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 · Zbl 1169.52004 [16] McMullen, P., Weights on polytopes, Discrete Comput. Geom., 15, 363-388, (1996) · Zbl 0849.52011 [17] Moon, J. W., Some enumerative results on series-parallel networks, Poznań, 1985, Amsterdam [18] Novik, I., The lower bound theorem for centrally symmetric simple polytopes, Mathematika, 46, 231-240, (1999) · Zbl 1037.52011 [19] Paffenholz, A.; Ziegler, G. M., The $$E\_{}\{t\}$$-construction for lattices, spheres and polytopes, Discrete Comput. Geom., 32, 601-624, (2004) · Zbl 1109.52009 [20] Roth, B., Rigid and flexible frameworks, Am. Math. Mon., 88, 6-21, (1981) · Zbl 0455.51012 [21] Sanyal, R.: Constructions and obstructions for extremal polytopes. Ph.D. thesis, TU Berlin (2008) · Zbl 1244.52003 [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 [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] Stanley, R., Generalized $$H$$-vectors, intersection cohomology of toric varieties, and related results, Kyoto, 1985, Amsterdam [25] Stanley, R., On the number of faces of centrally-symmetric simplicial polytopes, Graphs Comb., 3, 55-66, (1987) · Zbl 0611.52002 [26] Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. AMS, Providence (1996) · Zbl 0856.13020 [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] Whiteley, W., Infinitesimally rigid polyhedra. I. Statics of frameworks, Trans. Am. Math. Soc., 285, 431-465, (1984) · Zbl 0518.52010 [29] Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.