After successful completion of the course, students are able to address questions in convex geometry in independent research.  In particular, this creates prerequisites for the later writing of a master's thesis.

Convex cones play a central role in various areas of mathematics, ranging from optimization to algebraic geometry. The seminar is dedicated to the geometric properties of convex cones in Euclidean space. It is based on the recent book Convex Cones: Geometry and Probability by Rolf Schneider. It covers (among others) the following topics

• Angles of polyhedra as functionals of their normal/tangent cones
  They lead to the Brianchon-Gram-Sommerville relations, which are angle sum theorems for high-dimensional polyhedra.
• Intrinsic volumes and Grassman angles fo convex cones
  These are exciting variants of the established concepts for convex bodies. They offer a promising perspective on open questions in spherical geometry.
• Valuations on convex cones
  The main question here is the characterization of the conic intrinsic volumes as SO(d)-invariant valuations in the sense of the celebrated Hadwiger theorem.
• Central hyperplane arrangements
  Finitely many hyperplanes through the origin decompose the space in polyhedral cones. If these planes are chosen randomly, the integral geometry of the intrinsic volumes  of the resultung cones leads to interesting results in stochastic geometry.
• Co-convex bodies
  These are complements of certain convex sets within pointed cones that occur, for instance, in hyperbolic geometry. There is a natural addition of such sets, which leads to a strong Brunn-Minkowski inequality in this setting.

Suggested Topics from the book:

• Section 1.9 A characterization of polarity. This section essentially contains one difficult (but self-contained) proof that characterizes the polarity operations on non-planar cones as an inclusion-reversing involution. It would be awesome to summarize the proof so that the geometric intuition (if any?) becomes evident!
• Section 2.2 Angles. ... where the Brianchon-Gram-Sommerville relations are proven. It heavily builds on Sections 1.7 and 2.1 but, should be accessible without prior knowledge.
• Leo Brauner: Sections 2.3&2.4 Intrinsic volumes for polyhedral cones. It contains the definition of polyhedral intrinsic volumes and so-called Grassman-angles (dual to quermass integrals), as well as first relations among the intrisic volumes, their behaviour under polarity and also a Guass-Bonnet formular for polyhedra. Some intuition from the theory of polytopes might help, but it is understandable and interesting on its own. The intrinsic volumes are crucial for the rest of the book up to Chapter 7, so this topic should be covered by someone.
• Yinxiang Hu: Section 3.4 Inequalities in spherical space. Spherical isoperimetric and Blaschke-Santalo inequality. Quite geometric but not too technical.
• Yinxiang Hu: Section 3.3 Valuations on the sphere. Deals with the problem of characterizing intrinsic volumes as continuous SO(d)-invariant valuations on the class of convex cones. This is a very interesting and very open problem. The book contains a partial result in the form of Theorem 3.3.2. In a talk, it might be interesting to give some context beyond Schneider's book.
• Paul Strasser: Section 5.1 (Non-generic) central hyperplane arrangements. It uses the so-called characteristic polynomial, a combinatorial invariant, in order to describe not only the number of k-cones in a d-dimensional hyperplane arrangenment but also their intrinsic volumes. This section uses some poset theory and is best-suited for someone who enjoys combinatorics.
• Section 5.2 Absorption probabilities. Here the probability that a random process "absorbs" the origin is computed. The arguments are really nice and rather elementary. Maybe the section is a bit short on its own. One could spice up the talk by including asymptotic results from Section 6.5.
• Intersections of random cones. Uses the kinematic formular (Section 4.3) for spherically intrinsic volume in order to compute the probability that two random cones have a non-zero intersection. The kinematic formular gives results for the case where one cone is fixed and the other is a random rotation of a fixed cone. More sophisticated random models are treated in Sections 5.3 to 5.6. This is certainly enough for two speakers. It might be hard without prior knowledge in convex and/or integral geometry. 
• Oscar Ortega Moreno: Sections 6.3 to 6.4. High-dimensional phenomena. Deals with exciting threshold phenomena for random hyperplane arrangements as the dimension grows to infinity. 
• Jacopo Ulivelli: Chapter 7. Coconvex sets. Deals with the geometry of (complements of) convex sets within a pointed cone. A lot of questions from classical convex geometry carry over to this setting, such as  e.g. a Brunn-Minkowski type inequality for the volume (which is particularly difficult if the complements in question are unbounded), or a classification of surface area measures. This section can be done by more than 1 person. Background in convex geometry will be helpful.

Introduction to independent literature research and presentation of mathematical facts. Feedback on oral presentation. Individual meetings with the supervisor.

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