# 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

1. The Helly theorem
2. Carathéodory and Bárány theorems
3. The Borsuk conjecture
4. Fair division
5. Inscribed and circumscribed polygons
6. Dyson and Kakutani theorems
7. Geometric inequalities
8. Combinatorics of convex polytopes
9. Center of mass, billiards and the variational principle
10. Geodesics and quasi-geodesics
11. The Steinitz theorem and its extensions
12. Universality of point and line configurations
13. Universality of linkages
14. Triangulations
15. Hilbert's third problem
16. Polytope algebra
17. Dissections and valuations
18. Monge problem for polytopes
19. Regular polytopes
20. Kissing numbers
21. ### Part II. Discrete Geometry of Curves and Surfaces

22. The four vertex theorem
23. Relative geometry of convex polygons
24. Global invariants of curves
25. Geometry of space curves
26. Geometry of convex polyhedra: basic results
27. Cauchy theorem: the statement, the proof and the story
28. Cauchy theorem: extensions and generalizations
29. Mean curvature and Pogorelov's lemma
30. Senkin-Zalgaller's proof of the Cauchy theorem
31. Flexible polyhedra
32. The algebraic approach
33. Static rigidity
34. Infinitesimal rigidity
35. Proof of the bellows conjecture
36. The Alexandrov curvature theorem
37. The Minkowski theorem
38. The Alexandrov existence theorem
39. Bendable surfaces
40. Volume change under bending
41. 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

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Last updated 9/10/2009.