Remote Access Electronic Research Announcements

Electronic Research Announcements

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
DOI: https://doi.org/10.1090/S1079-6762-04-00137-4
Published electronically: December 1, 2004
MathSciNet review: 2119033
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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  $\mathbb{R} ^4$.


References [Enhancements On Off] (What's this?)

  • 1. 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, https://doi.org/10.1090/conm/223/03132
  • 2. David W. Barnette, Projections of 3-polytopes, Israel J. Math. 8 (1970), 304–308. MR 0262923, https://doi.org/10.1007/BF02771563
  • 3. M. M. BAYER, The extended $f$-vectors of $4$-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. 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, https://doi.org/10.1201/9780203911211.ch18
  • 6. G. Gévay, Kepler hypersolids, Intuitive geometry (Szeged, 1991) Colloq. Math. Soc. János Bolyai, vol. 63, North-Holland, Amsterdam, 1994, pp. 119–129. MR 1383617
  • 7. 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
  • 8. 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
  • 9. 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, https://doi.org/10.1007/s004540010039
  • 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 $E_t$-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.
  • 13. Ludwig Schläfli, Gesammelte mathematische Abhandlungen. Band I, Verlag Birkhäuser, Basel, 1950 (German). MR 0034587
  • 14. T. SCHRÖDER, On neighborly cubical spheres and polytopes. Work in progress, TU Berlin, 2004.
  • 15. R. P. STANLEY, Generalized $h$-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. Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028
  • 18. Tatsien Li (ed.), Proceedings of the International Congress of Mathematicians. Vol. III, Higher Education Press, Beijing, 2002. Invited lectures; Held in Beijing, August 20–28, 2002. MR 1957513
  • 19. -, 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.

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 52B05, 52B11, 52B12

Retrieve articles in all journals with MSC (2000): 52B05, 52B11, 52B12


Additional Information

Günter M. Ziegler
Affiliation: Inst. Mathematics, MA 6-2, TU Berlin, D-10623 Berlin, Germany
Email: ziegler@math.tu-berlin.de

DOI: https://doi.org/10.1090/S1079-6762-04-00137-4
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.

American Mathematical Society