Convex Geometry

Code: BMETE94AM22
Requirements: 2/2/0/V/4
Semester: 2025/26/2
Language: English
Instructor: Dr. Péter Vrana (E0 course)

Attendance requirements

Attendance is mandatory on at least 50% of lectures and 70% of exercise classes.

Midterm tests

For signature it is required – in addition to attendance requirements – to complete both midterm tests with at least 6 point scores after replacement tests. Both midterms can be replaced or improved. In case of replacement/improvement, the score of the replacement test overwrites the earlier result even if the score of the replacement test is less.

  • 1st midterm: March 27, Friday, from practice problems (45 minutes, 20 points)
  • 2nd midterm: May 15, Friday, from practice problems (45 minutes, 20 points)

Assignment

MSc and PhD studends are required to hand in the solutions by May 29. Please see the assignment problems here.

Exam

The final grade is determined based on an oral exam.

Schedule

  • February 20, Affine subspaces, affine combinations, convex sets, convex combinations
  • February 27, Convex hull, theorems of Radon and Carathéodory
  • March 6, Helly's theorem, hyperplanes, Jung's theorem
  • March 13, Minkowski addition, separation
  • March 20, Supporting hyperplanes, faces of a convex set, extremal and exposed points, the Krein–Milman theorem
  • March 27, 1st midterm, Algebra of convex sets, valuations. Euler characteristic
  • April 3, no class (Good Friday)
  • April 10, no class (Spring Holiday)
  • April 17, Polytopes, polyhedral sets, their face structures
  • April 24, Euler's theorem for polytopes
  • May 1, no class (Labour Day)
  • May 8, Polarity, duality theorem for polytopes
  • May 15, 2nd midterm, Distance of convex bodies: Hausdorff and Banach–Mazur distance
  • May 22, Ellipsoids
  • May 29, Cyclic polytopes

Lectures

The latest version of the lecture notes may be found here.

Exercises

Problem sheets, solutions.

Consultation

By appointment.

Recommended literature

  1. R. Tyrrell Rockafellar, Convex Analysis, Princeton University Press, Princeton NJ, 1972.
  2. Alexander Barvinok, A Course in Convexity, Graduate Studies in Mathematics 54, Amer. Math. Soc., Providence RI, 2002.
  3. Jiři Matoušek, Lectures in Discrete Geometry, Graduate Texts in Mathematics 212, Springer, New York, 2002.
  4. Branko Grünbaum, Convex polytopes, Graduate Texts in Mathematics 221, Springer, New York, 2003.