| Area of Mathematics: | Algebra and Geometry (EAG) | ||
| Semester: | 5ο | ||
| Course ID: | 52203 | ||
| Course Type: | Elective | ||
| Teaching hours per week: | Theory: 4 | Practice: 0 | Laboratory: 0 |
| ECTS : | 5 | ||
| Eclass: | |||
| Instructors: |
Description
- Linear and correlated subspaces. Convex sets, convex combinations, convex position.
- Theorems of Caratheodory, Helly, Radon. Applications to combinatorial geometry and approximation theory
- Geometry of Numbers: Grids, Minkowski’s first theorem, applications to number theory, successive minima, Minkowski’s second theorem.
- Convex polytopes, Voronoi diagrams, hyperplanes order.
- Convex functions.
- Convex sets: Metric projection. Supporting planes. Separation theorems. Duality. Support function and level function. Extreme and exposed points. The Minkowski-Krein-Milman theorem. Applications (Birkhoff polytope, permutation polytope, inequalities for graph eigenvalues).
- Convex bodies: Hausdorff metric. Blaschke’s selection theorem. Steiner symmetrization. Geometric inequalities. Volume in the n-dimensional Euclidean space. “Paradoxes” in large dimensions. Brunn-Minkowski inequality. Isoperimetric problems.
- Discrete geometry topics: Ramsey geometric theory (Erdos-Szekeres theorem), coincidence problems (Szemeredi-Trotter theorem), implantation of finite metric spaces in spaces with norm.
Bibliography
P.Gruber P, Convex and Discrete Geometry , Springer , 2007.
R.Webster , Convexity , Oxford University Press , 1994.
