Convex Geometry (Summer Semester 2013)
- Lecturer: Prof. Dr. Daniel Hug
- Classes: Lecture (0152800), Problem class (0152810)
- Weekly hours: 4+2
|Problem class:||Monday 15:45-17:15||1C-04|
|Lecturer||Prof. Dr. Daniel Hug|
|Office hours: Nach Vereinbarung.|
|Room 2.051 Kollegiengebäude Mathematik (20.30)|
|Email: email@example.com||Problem classes||Dr. Ines Ziebarth|
|Office hours: nach Vereinbarung|
|Room Allianz-Gebäude (05.20)|
Convexity is a fundamental notion 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 segment joining the two points is also contained in . 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 or sets of constant width.
This course is suited for everybody with a firm background in analysis and linear algebra.
Lecture Notes and Exercises
Lecture notes in English (by D. Hug and W. Weil) and exercises are to be found here.
- Gruber, Peter. Convex and Discrete Geometry, Grundlehren der mathematischen Wissenschaften, vol. 336, Springer, Berlin, 2007.
- Schneider, Rolf. Convex Bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.