Code: BMETE94AM22;
Requirements: 2/2/0/V/4;
Semester: 2023/24/2;
Language: English;
Instructor: Dr. Péter Vrana (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 22, Friday, from practice problems related to Sections 1-5 in the lecture notes (45 minutes, 20 points)
  • Sample first midterm
  • Solution of sample first midterm
• 2nd midterm: May 10, Friday, from practice problems (45 minutes, 20 points)
  • Sample second midterm
  • Solution of sample second midterm
• Replacement tests: May 24.

Assignment

Please see the assignment problems here. MSc and PhD studends are required to hand in the solutions by May 31.

Exam

The final grade is determined based on an oral exam.

Schedule

• February 16, Affine subspaces, affine combinations, convex sets, convex combinations
• February 23, Convex hull, theorems of Radon and Carathéodory
• March 1, Helly's theorem, hyperplanes, Jung's theorem
• March 8, Minkowski addition, separation
• March 15, no class, National day
• March 22, 1st midterm, Supporting hyperplanes, faces of a convex set, extremal and exposed points, the Krein–Milman theorem
• March 29, no class, Good Friday
• April 5, no class, Spring break
• April 12, Algebra of convex sets, valuations. Euler characteristic
• April 19, Polytopes, polyhedral sets, their face structures
• April 26, Euler's theorem for polytopes
• May 10, 2nd midterm, Polarity, duality theorem for polytopes
• May 17, Distance of convex bodies: Hausdorff and Banach–Mazur distance
• May 24, replacement tests, Cyclic polytopes

Lectures

The latest version of the lecture notes may be found here. (Updated: April 14)

Exercises

• Problem sheet 1 (Affine and convex combinations), solutions
• Problem sheet 2 (Convex hull, theorems of Radon and Carathéodory), solutions
• Problem sheet 3 (Helly's theorem), solutions
• Problem sheet 4 (Hyperplanes, Minkowski sum, separation), solutions
• Problem sheet 5 (Supporting hyperplanes, faces of convex sets, extremal and exposed points, the Krein–Milman Theorem), solutions
• Problem sheet 6 (Algebra of convex sets, the Euler characteristic), solutions
• Problem sheet 7 (Polytopes, polyhedral sets, their face structures), solutions
• Problem sheet 8 (Euler's theorem), 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.