Convex Polytopes: Extremal Constructions and fVector Shapes
Abstract
These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of fvectors, and extremal constructions. The first lecture treats 3dimensional polytopes; it includes a complete proof of the KoebeAndreevThurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at fvector shapes of very highdimensional polytopes. The third lecture explains a surprisingly simple construction for 2simple 2simplicial 4polytopes, which have symmetric fvectors. Lecture 4 sketches the geometry of the cone of fvectors for 4polytopes, and thus identifies the existence/construction of 4polytopes 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2004
 arXiv:
 arXiv:math/0411400
 Bibcode:
 2004math.....11400Z
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Combinatorics;
 5201;
 5202;
 52B05;
 52B11;
 52B12
 EPrint:
 73 pages, large file. Lecture Notes for PCMI Summer Course, Park City, Utah, 2004