# Lectures on Discrete and Polyhedral Geometry

## Igor Pak, UCLA

The latest version of the book is available here (3Mb).

If you print it on a color printer the figures will look nicer.

*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
- Triangulations
- 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

### Minor stats

- 40 sections, 240 subsections, +appendix
- about 425 interesting pages
- about 500 exercises, most with solutions
- about 270 figures
- about 550 references

Click here
to return to Igor Pak Home Page.

To e-mail me click
here and delete what is clearly not needed.

*Last updated 9/10/2009.*