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

  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

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Last updated 9/10/2009.