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ISSN 1079-6762

Projected products of polygons

Author(s): Günter M. Ziegler
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 122-134.
MSC (2000): Primary 52B05; Secondary 52B11, 52B12
Posted: December 1, 2004
MathSciNet review: 2119033
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Abstract | References | Similar articles | Additional information

Abstract: It is an open problem to characterize the cone of -vectors of -dimensional convex polytopes. The question whether the ``fatness'' of the -vector of a -polytope can be arbitrarily large is a key problem in this context.

Here we construct a -parameter family of -dimensional polytopes $\pi(P^{2r}_n)$ with extreme combinatorial structure. In this family, the ``fatness'' of the -vector gets arbitrarily close to ; an analogous invariant of the flag vector, the ``complexity,'' gets arbitrarily close to .

The polytopes are obtained from suitable deformed products of even polygons by a projection to $\mathbb{R} ^4$ .

References:

1.: N. AMENTA AND G. M. ZIEGLER, Deformed products and maximal shadows, Advances in Discrete and Computational Geometry (South Hadley, MA, 1996), B. Chazelle, J. E. Goodman, and R. Pollack, eds., vol. 223 of Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 1998, pp. 57-90. MR 1661377 (2000a:52019)
2.: D. W. BARNETTE, Projections of -polytopes, Israel J. Math. 8 (1970), pp. 304-308. MR 0262923 (41:7528)
3.: M. M. BAYER, The extended -vectors of -polytopes, J. Combinatorial Theory, Ser. A, 44 (1987), pp. 141-151. MR 0871395 (88b:52009)
4.: M. M. BAYER AND L. J. BILLERA, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Inventiones Math. 79 (1985), pp. 143-157. MR 0774533 (86f:52010b)
5.: D. EPPSTEIN, G. KUPERBERG, AND G. M. ZIEGLER, Fat -polytopes and fatter -spheres, Discrete Geometry: In honor of W. Kuperberg's 60th birthday, A. Bezdek, ed., vol. 253 of Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2003, pp. 239-265.
arXiv:math.CO/0204007. MR 2034720 (2004j:52009)
6.: G. G´EVAY, Kepler hypersolids, Intuitive Geometry (Szeged, 1991), vol. 63 of Colloq. Math. Soc. János Bolyai, Amsterdam, 1994, North-Holland, pp. 119-129. MR 1383617 (97a:52013)
7.: B. GRÜNBAUM, Convex Polytopes, vol. 221 of Graduate Texts in Math., Springer-Verlag, New York, 2003. Second edition edited by V. Kaibel, V. Klee and G. M. Ziegler (original edition: Interscience, London 1967). MR 1976856 (2004b:52001)
8.: A. H¨OPPNER AND G. M. ZIEGLER, A census of flag-vectors of -polytopes, in Polytopes -- Combinatorics and Computation, G. Kalai and G. M. Ziegler, eds., vol. 29 of DMV Seminars, Birkhäuser-Verlag, Basel, 2000, pp. 105-110. MR 1785294 (2001e:52026)
9.: M. JOSWIG AND G. M. ZIEGLER, Neighborly cubical polytopes, Discrete Comput. Geometry (Grünbaum Festschrift (G. Kalai, V. Klee, eds.)) 24 (2000), pp. 325-344.
arXiv:math.CO/9812033. MR 1758054 (2001f:52019)
10.: G. KALAI, Rigidity and the lower bound theorem, I, Inventiones Math. 88 (1987), pp. 125-151. MR 0877009 (88b:52014)
11.: A. PAFFENHOLZ, New polytopes from products. Preprint, TU Berlin, November 2004, 22 pages. arXiv:math.MG/0411092.
12.: A. PAFFENHOLZ AND G. M. ZIEGLER, The -construction for lattices, spheres and polytopes. Discrete Comput. Geometry (Billera Festschrift (M. Bayer, C. Lee, B. Sturmfels, eds.)), in print; published online August 23, 2004; arXiv:math.MG/0304492.
13.: L. SCHLÄFLI, Theorie der vielfachen Kontinuität, Denkschriften der Schweizerischen naturforschenden Gesellschaft, Vol. 38, pp. 1-237, Zürcher und Furrer, Zürich, 1901. Written in 1850-1852. Reprinted in: Ludwig Schläfli, 1814-1895, Gesammelte Mathematische Abhandlungen, Vol. I, Birkhäuser, Basel, 1950, pp. 167-387. MR 0034587 (11:611b)
14.: T. SCHRÖDER, On neighborly cubical spheres and polytopes. Work in progress, TU Berlin, 2004.
15.: R. P. STANLEY, Generalized -vectors, intersection cohomology of toric varieties, and related results, Commutative Algebra and Combinatorics, M. Nagata and H. Matsumura, eds., vol. 11 of Advanced Studies in Pure Mathematics, Kinokuniya, Tokyo, 1987, pp. 187-213. MR 0951205 (89f:52016)
16.: E. STEINITZ, Über die Eulerschen Polyederrelationen, Archiv für Mathematik und Physik 11 (1906), pp. 86-88.
17.: G. M. ZIEGLER, Lectures on Polytopes, vol. 152 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995. Revised edition, 1998; ``Updates, corrections, and more'' at www.math.tu-berlin.de/ ziegler. MR 1311028 (96a:52011)
18.: -, Face numbers of -polytopes and -spheres, Proceedings of the International Congress of Mathematicians (ICM 2002, Beijing), L. Tatsien, ed., vol. III, Higher Education Press, Beijing, 2002, pp. 625-634.
arXiv:math.MG/0208073. MR 1957513 (2003i:00010b)
19.: -, Convex polytopes: Extremal constructions and -vector shapes. Park City Mathematical Institute (PCMI 2004) Lecture Notes. With an Appendix by Th. Schröder and N. Witte, 2004. Preprint, TU Berlin, November 2004, 73 pages.

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