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MATH 669: COMBINATORICS OF POLYTOPES
Alexander Barvinok
Abstract. These are rather condensed notes, not really proofread or edited, presenting key definitions and results of the course that I taught in Winter 2010 term.
Problems marked by are easy and basic, problems marked by
∗ may be difficult.
1. Basic definitions 3
2. Carath´eodory’s Theorem 4
3. Radon’s Theorem 5
4. Helly’s Theorem 6
5. Euler characteristic 7
6. Polyhedra and linear transformations 10
7. Minkowski sum 12
8. Examples of valuations 14
9. The structure of polyhedra 16
10. The Euler-Poincar´e formula 23
11. The Birkhoff polytope 24
12. The Schur-Horn Theorem 26
13. Transportation polyhedra 27
14. The permutation polytope 28
15. Cyclic polytopes 31
16. Polarity 32
17. Polytopes and polarity 36
18. Regular triangulations and subdivisions 38
19. The secondary polytope 39
20. Fiber polytopes 43
21. Identities modulo polyhedra with lines 46
22. The exponential valuation 49
23. A formula for the volume of a polytope 54
24. Simple polytopes and their h-vectors 55
25. The Upper Bound Theorem 58
26. Balinski’s Theorem 61
27. Reconstructing a simple polytope from its graph 63
28. The diameter of the graph of a polyhedron 65
29. Edges of a centrally symmetric polytope 66
30. Approximating a convex body by an ellipsoid 68
31. Spherical caps 70
32. An inequality for the number of faces of a centrally symmetric polytope 72
33. Gale transforms and symmetric Gale transforms 75
34. Almost Euclidean subspaces of ℓ
and centrally symmetric polytopes
with many faces 77
35. The volume ratio and almost Euclidean subspaces
MATH 669: COMBINATORICS OF POLYTOPES
Alexander Barvinok
Abstract. These are rather condensed notes, not really proofread or edited, presenting key definitions and results of the course that I taught in Winter 2010 term.
Problems marked by are easy and basic, problems marked by
∗ may be difficult.
1. Basic definitions 3
2. Carath´eodory’s Theorem 4
3. Radon’s Theorem 5
4. Helly’s Theorem 6
5. Euler characteristic 7
6. Polyhedra and linear transformations 10
7. Minkowski sum 12
8. Examples of valuations 14
9. The structure of polyhedra 16
10. The Euler-Poincar´e formula 23
11. The Birkhoff polytope 24
12. The Schur-Horn Theorem 26
13. Transportation polyhedra 27
14. The permutation polytope 28
15. Cyclic polytopes 31
16. Polarity 32
17. Polytopes and polarity 36
18. Regular triangulations and subdivisions 38
19. The secondary polytope 39
20. Fiber polytopes 43
21. Identities modulo polyhedra with lines 46
22. The exponential valuation 49
23. A formula for the volume of a polytope 54
24. Simple polytopes and their h-vectors 55
25. The Upper Bound Theorem 58
26. Balinski’s Theorem 61
27. Reconstructing a simple polytope from its graph 63
28. The diameter of the graph of a polyhedron 65
29. Edges of a centrally symmetric polytope 66
30. Approximating a convex body by an ellipsoid 68
31. Spherical caps 70
32. An inequality for the number of faces of a centrally symmetric polytope 72
33. Gale transforms and symmetric Gale transforms 75
34. Almost Euclidean subspaces of ℓ
and centrally symmetric polytopes
with many faces 77
35. The volume ratio and almost Euclidean subspaces
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