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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: 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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Additional Information

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

DOI: http://dx.doi.org/10.1090/S1079-6762-04-00137-4
PII: S 1079-6762(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.