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Projected products of polygons
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## Projected products of polygons

### Günter M. Ziegler

**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~$16$.
\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)

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