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Convex Polytopes: Extremal constructions and f-vector shapes

Günter M. Ziegler – 2007

These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f-vectors. Lecture 4 sketches the geometry of the cone of f-vectors for 4-polytopes, and thus identifies the existence/construction of 4-polytopes of high ``fatness'' as a key problem. In this direction, the last lecture presents a very recent construction of ``projected products of polygons,'' whose fatness reaches 9-\eps.

Convex Polytopes: Extremal constructions and f-vector shapes
Günter M. Ziegler
Amer. Math. Society
Providence, RI
Erschienen in
"Geometric Combinatorics'', Proc. Park City Mathematical Institute (PCMI) 2004, pages 617-691