Convex Polytopes: Extremal Constructions and f-Vector Shapes

Abstract

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.

73 pages, large file. Lecture Notes for PCMI Summer Course, Park City, Utah, 2004; revised and slightly updated final version, December 2005

References in corpus (4)

• Geometric Reasoning with polymake
• A unique representation of polyhedral types
• Constructions for 4-Polytopes and the Cone of Flag Vectors
• Unimodality and convexity of f-vectors of polytopes

Cited by in corpus (10)

• Scribability problems for polytopes
• The geometry of high-dimensional phase diagrams: I. Generalized Gibbs Phase Rule
• A Taxonomy of Crystallographic Sphere Packings
• On face numbers of neighborly cubical polytopes
• Unimodality of f-vectors of cyclic polytopes
• Log-concavity of face vectors of cyclic and ordinary polytopes
• Kronecker covers, V-construction, unit-distance graphs and isometric point-circle configurations
• The Unimodality Conjecture for cubical polytopes
• Computing Circle Packing Representations of Planar Graphs
• A polytopal generalization of Apollonian packings and Descartes' theorem