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Institute of Stochastics

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 2.056 und 2.002

Address
Hausadresse:
Karlsruher Institut für Technologie (KIT)
Institut für Stochastik
Englerstr. 2
D-76131 Karlsruhe

Postadresse:
Karlsruher Institut für Technologie (KIT)
Institut für Stochastik
Postfach 6980
D-76049 Karlsruhe

Office hours:
Mo-Fr 10:00 - 12:00

Tel.: +49 721 608 43270/43265

Fax.: +49 721 608 46066

Convex Geometry (Summer Semester 2013)

Lecturer: Prof. Dr. Daniel Hug
Classes: Lecture (0152800), Problem class (0152810)
Weekly hours: 4+2


Schedule
Lecture: Tuesday 14:00-15:30 1C-04
Wednesday 11:30-13:00 1C-04
Problem class: Monday 15:45-17:15 1C-04
Lecturers
Lecturer Prof. Dr. Daniel Hug
Office hours: Nach Vereinbarung.
Room 2.051 Kollegiengebäude Mathematik (20.30)
Email: daniel.hug@kit.edu
Problem classes Dr. Ines Ziebarth
Office hours: nach Vereinbarung
Room Allianz-Gebäude (05.20)
Email: ines.ziebarth@kit.edu

Course description

Convexity is a fundamental notion in mathematics which has a combinatorial, an analytic, a geometric and a probabilistic flavour. Basically, a given set A in a real vector space is called convex if with any two points of A the segment joining the two points is also contained in A. 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:

  1. Geometric foundations: combinatorial properties, support and separation theorems, extremal representations
  2. Convex functions
  3. The Brunn-Minkowski Theory: basic functionals of convex bodies, mixed volumes, geometric (isoperimetric) inequalities
  4. Surface area measures and projection functions
  5. Integral geometric formulas

If time permits, we also consider additional topics such as symmetrization of convex sets or sets of constant width.


Prerequisites

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.


References

  • 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.