Convex Polytopes: Extremal Constructions and f-Vector Shapes

Details Convex Polytopes: Extremal Constructions and f-Vector Shapes Convex Polytopes: Extremal Constructions and f-Vector Shapes

2004-11-18

arxiv.org/abs/math/0411400v2

Mathematics

Metric Geometry

73 pages, large file. Lecture Notes for PCMI Summer Course, Park City, Utah, 2004; revised and slightly updated final version,

Scientific paper

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.

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