Lectures on Discrete and Polyhedral Geometry
Igor Pak, UCLA
The latest version of the book is available here (3Mb).
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Warning! This version is full of typos and math errors (hopefully, only minor).
I am slowly editing it them out. Please, please, let me know if you find some.
Table of Contents
Part I. Basic Discrete Geometry
- The Helly theorem
- Carathéodory and Bárány theorems
- The Borsuk conjecture
- Fair division
- Inscribed and circumscribed polygons
- Dyson and Kakutani theorems
- Geometric inequalities
- Combinatorics of convex polytopes
- Center of mass, billiards and the variational principle
- Geodesics and quasi-geodesics
- The Steinitz theorem and its extensions
- Universality of point and line configurations
- Universality of linkages
- Hilbert's third problem
- Polytope algebra
- Dissections and valuations
- Monge problem for polytopes
- Regular polytopes
- Kissing numbers
Part II. Discrete Geometry of Curves and Surfaces
- The four vertex theorem
- Relative geometry of convex polygons
- Global invariants of curves
- Geometry of space curves
- Geometry of convex polyhedra: basic results
- Cauchy theorem: the statement, the proof and the story
- Cauchy theorem: extensions and generalizations
- Mean curvature and Pogorelov's lemma
- Senkin-Zalgaller's proof of the Cauchy theorem
- Flexible polyhedra
- The algebraic approach
- Static rigidity
- Infinitesimal rigidity
- Proof of the bellows conjecture
- The Alexandrov curvature theorem
- The Minkowski theorem
- The Alexandrov existence theorem
- Bendable surfaces
- Volume change under bending
- Foldings and unfoldings
- 40 sections, 240 subsections, +appendix
- about 425 interesting pages
- about 500 exercises, most with solutions
- about 270 figures
- about 550 references
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Last updated 9/10/2009.