Projected products of polygons

Ziegler, Günter

doi:10.1090/S1079-6762-04-00137-4

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 $\mathbb {R}^4$.

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