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
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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 $\mathbb {R}^4$.
- 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 10.1090/conm/223/03132
- David W. Barnette, Projections of 3-polytopes, Israel J. Math. 8 (1970), 304–308. MR 262923, DOI 10.1007/BF02771563
- Margaret Bayer, The extended $f$-vectors of $4$-polytopes, J. Combin. Theory Ser. A 44 (1987), no. 1, 141–151. MR 871395, DOI 10.1016/0097-3165(87)90066-5
- 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 10.1090/conm/034/777703
- 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 10.1201/9780203911211.ch18
- 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, DOI 10.1007/978-1-4613-0019-9
- 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 10.1007/s004540010039
- Gil Kalai, Rigidity and the lower bound theorem. I, Invent. Math. 88 (1987), no. 1, 125–151. MR 877009, DOI 10.1007/BF01405094
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, DOI 10.1007/978-3-0348-4118-4
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 10.2969/aspm/01110187
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, DOI 10.1007/978-1-4613-8431-1
- 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.
Z51a 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.
Bar4 D. W. Barnette, Projections of $3$-polytopes, Israel J. Math. 8 (1970), pp. 304–308.
Bay M. M. Bayer, The extended $f$-vectors of $4$-polytopes, J. Combinatorial Theory, Ser. A, 44 (1987), pp. 141–151.
BaBi 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.
Z80 D. Eppstein, G. Kuperberg, and G. M. Ziegler, Fat $4$-polytopes and fatter $3$-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.
Gevay G. Gévay, Kepler hypersolids, Intuitive Geometry (Szeged, 1991), vol. 63 of Colloq. Math. Soc. János Bolyai, Amsterdam, 1994, North-Holland, pp. 119–129.
Gr1-2 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).
Z59 A. Höppner and G. M. Ziegler, A census of flag-vectors of $4$-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.
Z62 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.
kalai87:rigidi G. Kalai, Rigidity and the lower bound theorem, I, Inventiones Math. 88 (1987), pp. 125–151.
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.
Schla 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.
schroeder04 T. Schröder, On neighborly cubical spheres and polytopes. Work in progress, TU Berlin, 2004.
Sta7 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.
Stei3 E. Steinitz, Über die Eulerschen Polyederrelationen, Archiv für Mathematik und Physik 11 (1906), pp. 86–88.
Z35 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.
Z82 —, Face numbers of $4$-polytopes and $3$-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.
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
Email:
ziegler@math.tu-berlin.de
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.
