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

     Part II. Discrete Geometry of Curves and Surfaces

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