Convex Geometry (Summer Semester 2021)
- Lecturer: Prof. Dr. Daniel Hug
- Classes: Lecture (0152800), Problem class (0152810)
- Weekly hours: 4+2
All relevant information and documents will be provided via the ILIAS-System.
| Schedule | ||
|---|---|---|
| Lecture: | Wednesday 12:00-13:30 | 20.30 SR 2.58 |
| Thursday 12:00-13:30 | 20.30 SR 3.69 | |
| Problem class: | Tuesday 8:00-9:30 | 20.30 SR 2.58 |
| Lecturers | ||
|---|---|---|
| Lecturer | Prof. Dr. Daniel Hug | |
| Office hours: Nach Vereinbarung. | ||
| Room 2.051 Kollegiengebäude Mathematik (20.30) | ||
| Email: daniel.hug@kit.edu | Problem classes | Dominik Pabst M.Sc. |
| Office hours: by appointment | ||
| Room 2.008 Kollegiengebäude Mathematik (20.30) | ||
| Email: dominik.pabst@kit.edu | ||
Topics
Convexity is a fundamental concept in mathematics which has a combinatorial, an analytic, a geometric and a probabilistic flavour. Basically, a given set in a real vector space is called convex if with any two points of the set also the segment joining the two points is contained in the set. This course provides an introduction to the geometry of convex sets in a finite-dimensional real vector space and to basic properties convex functions. Results and methods of convex geometry are particularly relevant, for instance, in optimization theory and in stochastic geometry.
The following topics will be covered:
- Geometric foundations: combinatorial properties, support and separation theorems, extremal representations
- Convex functions
- The Brunn-Minkowski Theory: basic functionals of convex bodies, mixed volumes, geometric (isoperimetric) inequalities
- Surface area measures and projection functions
- Integral geometric formulas
If time permits, we also consider additional topics such as symmetrization of convex sets.
Prerequisits
Linear Algebra I + II, Analysis I - III. Some basic facts of functional analysis will be provided as the need arises.
Examination
Oral examinations by individual appointment
References
• Hug, D., Weil, W. Lectures on Convex Geometry, GTM 286, Springer, Cham, 2020.
https://link.springer.com/book/10.1007%2F978-3-030-50180-8
• Schneider, R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, 2014.
• Webster, R. Convexity. Oxford University Press, 1994.
• Gruber, P.M. Convex and Discrete Geometry. Grundlehren 336, Springer, 2007.