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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~$16$. \par The polytopes are obtained from suitable deformed products of even polygons by a projection to~$\mathbb{R}^4$.

Copyright 2004 American Mathematical Society
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Article Info

• ERA Amer. Math. Soc. 10 (2004), pp. 122-134
• Publisher Identifier: S 1079-6762(04)00137-4
• 2000 Mathematics Subject Classification. Primary 52B05; Secondary 52B11, 52B12
• Key words and phrases. Discrete geometry, convex polytopes, $f$-vectors, deformed products of polygons
• Received by editors July 4, 2004
• Received by editors in revised form September 17, 2004
• Posted on December 1, 2004
• Communicated by Sergey Fomin
• Comments (When Available)

Günter M. Ziegler

Inst. Mathematics, MA 6-2, TU Berlin, D-10623 Berlin, Germany

E-mail address: ziegler@math.tu-berlin.de

Partially supported by Deutsche Forschungs-Gemeinschaft (DFG), via the \emph{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)