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ISSN 1079-6762



Projected products of polygons

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


It is an open problem to characterize the cone of $f$-vectors of $4$-dimensional convex polytopes. The question whether the “fatness” of the $f$-vector of a $4$-polytope can be arbitrarily large is a key problem in this context. Here we construct a $2$-parameter family of $4$-dimensional polytopes $\pi (P^{2r}_n)$ with extreme combinatorial structure. In this family, the “fatness” of the $f$-vector gets arbitrarily close to $9$; an analogous invariant of the flag vector, the “complexity,” gets arbitrarily close to $16$.

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

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  • Nina Amenta and Günter M. Ziegler, Deformed products and maximal shadows of polytopes, Advances in discrete and computational geometry (South Hadley, MA, 1996) Contemp. Math., vol. 223, Amer. Math. Soc., Providence, RI, 1999, pp. 57–90. MR 1661377, DOI
  • David W. Barnette, Projections of 3-polytopes, Israel J. Math. 8 (1970), 304–308. MR 262923, DOI
  • Margaret Bayer, The extended $f$-vectors of $4$-polytopes, J. Combin. Theory Ser. A 44 (1987), no. 1, 141–151. MR 871395, DOI
  • Margaret M. Bayer and Louis J. Billera, Counting faces and chains in polytopes and posets, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 207–252. MR 777703, DOI
  • David Eppstein, Greg Kuperberg, and Günter M. Ziegler, Fat 4-polytopes and fatter 3-spheres, Discrete geometry, Monogr. Textbooks Pure Appl. Math., vol. 253, Dekker, New York, 2003, pp. 239–265. MR 2034720, DOI
  • G. Gévay, Kepler hypersolids, Intuitive geometry (Szeged, 1991) Colloq. Math. Soc. János Bolyai, vol. 63, North-Holland, Amsterdam, 1994, pp. 119–129. MR 1383617
  • Branko Grünbaum, Convex polytopes, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR 1976856
  • Andrea Höppner and Günter M. Ziegler, A census of flag-vectors of 4-polytopes, Polytopes—combinatorics and computation (Oberwolfach, 1997) DMV Sem., vol. 29, Birkhäuser, Basel, 2000, pp. 105–110. MR 1785294
  • M. Joswig and G. M. Ziegler, Neighborly cubical polytopes, Discrete Comput. Geom. 24 (2000), no. 2-3, 325–344. The Branko Grünbaum birthday issue. MR 1758054, DOI
  • Gil Kalai, Rigidity and the lower bound theorem. I, Invent. Math. 88 (1987), no. 1, 125–151. MR 877009, DOI
  • paffenholz-pc A. Paffenholz, New polytopes from products. Preprint, TU Berlin, November 2004, 22 pages. arXiv:math.MG/0411092. Z89 A. Paffenholz and G. M. Ziegler, The $E_t$-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.
  • Ludwig Schläfli, Gesammelte mathematische Abhandlungen. Band I, Verlag Birkhäuser, Basel, 1950 (German). MR 0034587
  • schroeder04 T. Schröder, On neighborly cubical spheres and polytopes. Work in progress, TU Berlin, 2004.
  • Richard Stanley, Generalized $H$-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 187–213. MR 951205, DOI
  • Stei3 E. Steinitz, Über die Eulerschen Polyederrelationen, Archiv für Mathematik und Physik 11 (1906), pp. 86–88.
  • Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028
  • Tatsien Li (ed.), Proceedings of the International Congress of Mathematicians. Vol. II, Higher Education Press, Beijing, 2002. Invited lectures; Held in Beijing, August 20–28, 2002. MR 1957512
  • Z99 ---, Convex polytopes: Extremal constructions and $f$-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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Additional Information

Günter M. Ziegler
Affiliation: Inst. Mathematics, MA 6-2, TU Berlin, D-10623 Berlin, Germany

Keywords: Discrete geometry, convex polytopes, $f$-vectors, deformed products of polygons
Received by editor(s): July 4, 2004
Published electronically: December 1, 2004
Additional Notes: Partially supported by Deutsche Forschungs-Gemeinschaft (DFG), via the Matheon Research Center “Mathematics for Key Technologies” (FZT86), the Research Group “Algorithms, Structure, Randomness” (Project ZI 475/3), and a Leibniz grant (ZI 475/4)
Communicated by: Sergey Fomin
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.