Convex Geometry (Wintersemester 2008/09)
- Dozent*in: Prof. Dr. Daniel Hug
- Veranstaltungen: Vorlesung (1044), Übung (1045)
- Semesterwochenstunden: 4+2
| Termine | ||
|---|---|---|
| Vorlesung: | Montag 11:30-13:00 | Seminarraum 34 |
| Dienstag 11:30-13:00 | Seminarraum 34 | |
| Übung: | Mittwoch 14:00-15:30 | Seminarraum 31 |
| Lehrende | ||
|---|---|---|
| Dozent | Prof. Dr. Daniel Hug | |
| Sprechstunde: Nach Vereinbarung. | ||
| Zimmer 2.051 Kollegiengebäude Mathematik (20.30) | ||
| Email: daniel.hug@kit.edu | Übungsleiter | Dr. Mario Hörig |
| Sprechstunde: | ||
| Zimmer Allianz-Gebäude (05.20) | ||
| Email: Hoerig@math.uni-karlsruhe.de | ||
Inhalt
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.
Lecture notes in English will be available.
Details:
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
- Integral geometric formulas
If time permits, we also consider additional topics such as symmetrization of convex sets or sets of constant width.
Voraussetzungen
This course is suited for everybody with a firm background in analysis and linear algebra.
Übungsaufgaben
- Übungsblatt 1: sheet01.pdf
- Übungsblatt 2: sheet02.pdf including corrected definition in ex. 7
- Übungsblatt 3: sheet03.pdf
- Übungsblatt 4: sheet04.pdf
- Übungsblatt 5: sheet05.pdf
- Übungsblatt 6: sheet06.pdf
- Übungsblatt 7: sheet07.pdf
- Übungsblatt 8: sheet08.pdf
- Übungsblatt 9: sheet09.pdf
- Übungsblatt 10: sheet10.pdf
- Übungsblatt 11: sheet11.pdf
- Übungsblatt 12: sheet12.pdf including correction in ex. 45
- Übungsblatt 13: sheet13.pdf
Literaturhinweise
- 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.