**Week 1 (February 12) **Affine subspaces, affine combinations, convex sets, convex combinations.**Week 2 (February 19)** Convex hull, theorems of Radon and Carathéodory.**Week 3 (February 26) **Helly's theorem, hyperplanes, linear functionals.**Week 4 (March 5) **Minkowski addition, separation.**Week 5 (March 12) **Supporting hyperplanes, faces of a convex set, extremal and exposed points, the Krein-Milman theorem.**Week 6 (March 19)** Algebra of convex sets, valuations.**Week 7 (March 26) 1st midterm, **Euler characteristic.**Week 8 (April 2) **Good Friday, Spring break.**Week 9 (April 9) **Polytopes, polyhedral sets, their face structures.**Week 10 (April 16)** Euler's theorem for polytopes.**Week 11 (April 23) **Polarity, duality theorem for polytopes.**Week 12 (April 30) 2nd midterm, **a special class of polytopes: cyclic polytopes.**Week 13 (May 7) **Distance of convex bodies: Hausdorff and Banach-Mazur distance.**Week 14 (May 14) **Ellipsoids, the Löwner-John ellipsoid.

Problem sheets and other information

**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.