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# V.A.Timorin

## Convex sets

## Lecture notes

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## Course syllabus

- 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)

- 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

- 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

- 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