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V.A.Timorin

Convex sets

Lecture notes

Gzipped postscript (may be viewed directly by some versions of Ghostview)

[Lecture 1 (711K)|Lecture 2 (37K)|Lecture 3 (67K)|Lecture 4 (43K)
Lecture 5 (33K)|Lecture 6 (40K)|Lecture 7 (56K)|Lecture 8 (40K)
Lecture 9 (39K)]

Zipped postscript

[Lecture 1 (71K)|Lecture 2 (38K)|Lecture 3 (67K)|Lecture 4 (43K)
Lecture 5 (33K)|Lecture 6 (41K)|Lecture 7 (56K)|Lecture 8 (40K)
Lecture 9 (39K)]

Course syllabus

1. Geometric inequalities.
   • Isoperimetric inequality
   • The Steiner symmetrization
   • The Minkowski addition
   • The Hausdorff metric
   • Kugelungsatz von Blaschke
   • The Brunn-Minkowski inequality
   • Mixed volumes
   • The Aleksandrov-Fenchel inequality (without proof)
2. Topology of convex sets
   • Convex hull, the Caratheodory theorem
   • The Radon theorem
   • Separation conditions, the Hahn-Banach theorem
   • Support function, the Minkowski functional, duality
   • The Krein-Milman theorem
   • The Helly theorem
   • Convex polytopes, duality
   • The Weil-Minkowski theorem
   • Fans, dual fan of convex polytope
3. Combinatorics of convex polytopes
   • Simple and simplicial polytopes
   • Combinatorial equivalence of polytopes
   • Cyclic polytopes
   • f-vector and h-vector of simple (simplicial) polytope
   • The Euler theorem, the Dehn-Sommerville relations
   • Combinatorics of hyperplane sections
   • Gale diagrams
   • McMullen conditions (without proof), Upper Bound and Lower Bound theorems
   • The Billera-Lee theorem
   • The combinatorial theorem of Macaulay
4. Theory of volumes of simple convex polytopes
   • Support numbers
   • Formulae for volumes and mixed volumes
   • The polytope algebra
   • The Minkowski theorem
   • Proof of the Aleskandrov-Fenchel theorem for simple convex polytopes
   • The Aleksandrov inequality on mixed discriminants
   • The van der Waerden conjecture
   • Newton polytopes and the Bernstein theorem