This course (in English, as the first part of a three semester sequence) will be an introduction to the theory of Convex Polytopes, a core field of Discrete Geometry. The lectures will

  • introduce basic facts about polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin eliminiation, Schlegel diagrams, shellability, Gale diagrams, and oriented matroids
  • discuss important example and elegant constructions (cyclic and neighborly polytopes, zonotopes, Minkowski sums, permuathedra and associahedra, fiber polytopes, and the Lawrence construction)
  • show the excitement of current work in the field.

The course will be based on Günter M. Ziegler's 1995 text book "Lectures on Polytopes". This book has quickly become a standard text book for this field - but it also has triggered a lot of research, many of the open problems from 1995 have been solved by now, so the book does not any more cover "the excitement of current work in the field" - however, the course will try to provide this. In parallel to the course we will revise and update the book, towards a 2016 second edition of "Lectures on Polytopes". You are invited to get involved in this: By criticizing the old text, suggesting improvements and additions, researching new references, contributing problems, etc.

Here is the list of chapters of the book, and thus (roughly) the course contents:

  1. Introduction and Examples
  2. Polytopes, Polyhedra, and Cones
  3. Faces of Polytopes
  4. Graphs of Polytopes
  5. Steinitz' Theorem for 3-Polytopes
  6. Schlegel Diagrams for 4-Polytopes
  7. Duality, Gale Diagrams, and Applications
  8. Fans, Arrangements, Zonotopes, and Tilings
  9. Shellability and the Upper Bound Theorem
  10. Fiber Polytopes, and Beyond



  • Branko Grünbaum: "Convex Polytopes" (Original edition: Interscience 1967; 2nd edition, Graduate Texts in Math. 221, Springer 2003)
  • Peter McMullen, Geoffrey C. Shephard: "Convex Polytopes and the Upper Bound Conjecture" (London Math. Soc. Lecture Notes Series 3, Cambridge 1971)

  • Richard P. Stanley: "Enumerative Combinatorics", Vol.1 (Cambridge 1997)

  • Günter M. Ziegler "Lectures in Polytopes" (Graduate Texts in Math. 152, 1995; 7th revised printing 2007)


Additional Information:

Solid background in linear algebra. Knowledge in combinatorics and geometry is advantageous.


In the tutorial, we will occasionally review topics from the lecture but mostly discuss problems and solutions to homework assignments. You are encouraged to pitch in by presenting a solution every once in a while.

The problem sheets will usually be uploaded on Wednesdays on the course homepage under "Resources" and solutions should be turned in at the beginning of the exercise session on the following Wednesday.

You will receive points for your solutions based on whether your solutions are correct and well-written.

Submit your homework assignment in pairs, and indicate for each solution who wrote it up.

Every student is required to write up solutions to at least 40% out of all problems submitted throughout the semester.

Extra credit: +5 points (once) for presenting at least one correct solution at the blackboard during the exercise sessions.


Course requirements are the following:

(1) You must score at least 50% of the total points of the problems assigned, 130 points in total.

There will be 13 (+1) problem sheets and a sheet will have problems worth 20 points.

(2) You must pass an exam at the end of the semester which alone will determine your grade.




final exam (written): Thu, 23 February, 2017

time and place: 2:00 p.m.-3:30 p.m., room T003 (lecture hall Takustr. 9)

make-up exam: Thu, 6 April, 2017 2:00 p.m., SR 031/A6

If you would like to take an oral exam instead of the written one, please write an e-mail to Prof. Blagojević and Hannah Sjöberg by February 16, 2017. Note that the decision is binding. The type of examination (written/oral) of the make-up exam is the same as in the ordinary exam.