## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, April 13, 2004, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Andreas Paffenholz (TU Berlin)

## New Polytopes derived from Products

Recently J. Bokowski and G. Gevay presented a new family S_{mn} of self-dual 3-spheres based on a construction of G. Gevay. This family contains in particular the hypersimplex and the 24-cell.

In my talk I will show that this family of spheres is a very special case of the 'E-construction'. This method was invented by D. Eppstein, G. Kuperberg and G. M. Ziegler for the construction of 2-simple and 2-simplicial 4-polytopes and subsequently extended to arbitrary polytopes and, more general, to lattices by G. M. Ziegler and myself.

The family of spheres S_{mn} can be derived from products (C_m x C_n) of two polygons C_m and C_n with m resp. n vertices. We prove that all these spheres are in fact polytopal with flexible geometric realizations. For 1/m+1/n>=1/2 we obtain symmetric'' realizations of these polytopes, while such realizations cannot exist for other m,n. We will in particular discuss the realization space of the hypersimplex and the 24-cell.

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