Convex Geometry

Code: BMETE94AM22;
Requirements: 2/2/0/V/4;
Semester: 2024/25/2;
Language: English;
Instructor: Dr. Zsolt Lángi (E0 course)

Attendance requirements

In offline education attendance is mandatory on at least 50% of lectures and 70% of tutorials.

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 21, Friday, from practice problems related to the topics of the first five weeks (45 minutes, 20 points)
  • 2nd midterm: May 9, Friday, from practice problems related to the topics covered on the 6th-11th weeks (45 minutes, 20 points)
  • Replacement tests: . You may retake any or both of the midterm exams if you missed or failed the first occasion or wish to improve the points.
  • MSc and PhD studends are required to hand in the solutions of an assignment by May 30.

Exam

The final grade is determined based on an oral exam.

Schedule

  • February 15, Affine subspaces, affine combinations, convex sets, convex combinations.
  • February 22, Convex hull, theorems of Radon and Carathéodory.
  • February 28, Helly's theorem, hyperplanes, Jung's theorem.
  • March 7, Minkowski addition, separation.
  • March 14, Supporting hyperplanes, faces of a convex set, extremal and exposed points, the Krein–Milman theorem.
  • March 22, 1st midterm, Algebra of convex sets, valuations.
  • March 28, Euler characteristic, polytopes.
  • April 4, Polytopes (continued), polyhedral sets, their face structures.
  • April 11,   Euler's theorem for polytopes.
  • April 18, no class, Good Friday.
  • April 25, no class, Spring break.
  • May 2, no class, holiday (class is moved to May 17).
  • May 9, 2nd midterm, Polarity, duality theorem for polytopes.
  • May 16,  Distance of convex bodies: Hausdorff and Banach–Mazur distance.
  • May 17, Cyclic polytopes.
  • May 23, Ellipsoids.

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.

More information can be found in the Teams group  "Convex Geometry 2024/25, Spring term" .