Realization Spaces of Polytopes

Winter Semester 2020/2021, VL 2

Marta Panizzut, Institut für Mathematik, TU Berlin

Registration via email is mandatory in order to participate. In the email please give your name, your affiliation, and indicate whether you wish to receive credits.

Vorlesung:Tuesday, 16:00-18:00
Be aware of the change in the schedule.

The course will be held online. The zoom link for the lectures will be shared with the registered participants.
The first lecture will take place on Tuesday, November 3rd.

Prerequisites: Diskrete Geometrie I and Algebra I.


The study of realization spaces of polytopes is a classical topic in geometric combinatorics at the interface between discrete and real algebraic geometry. We will begin with classical constructions and results, and then focus on recent advances.
Topics of the course are:
  • Main definitions and constructions
  • Three-dimensional vs higher dimesional polytopes
  • Comparison of different models
  • Recent advances


  • Gouveia, Macchia, Thomas, and Wiebe: The slack realization space of a polytope. SIAM J. Discrete Math. 2019
  • Gouveia, Macchia, and Wiebe: Combining realization space models of polytopes. arXiv. 2020
  • Rastanawi, Sinn, Ziegler: On the Dimensions of the Realization Spaces of Polytopes. arXiv. 2020
  • Richter-Gebert: Realization Spaces of Polytopes. Lecture Notes in Mathematics. Springer. 1996
  • Ziegler: Lectures on Polytopes. Graduate Texts in Mathematics. Springer. 2007
  • Ziegler: Nonrational configurations, polytopes and surfaces. The Mathematical Intelligencer 30. 2008


    The oral exams will take place at the end of February. Precise dates will be discussed individually with the students registered for credits.


  • November 03: Introduction to the course, Slides
  • November 10: Semialgebraic sets, Slides
  • November 17: Polarity, Slides
  • November 24: Steinitz Theorem, I, Slides
  • December 01: Steinitz Theorem, II, Slides
  • December 08: Realization spaces of 3-poytopes, Slides
  • December 15: Lawrence extensions and Shor normal forms, Slides
  • January 05: Building blocks, Slides
  • January 12: Encoding semialgebraic sets in polytopes, Slides
  • January 19: Universality Theorem for 4-dimensional polytopes, Slides
  • January 26: Salck realization spaces of polytopes, I Slides
  • February 02: Slack realization spaces of polytopes, II Slides
  • February 09: No Lecture.
  • February 16: No Lecture.
  • February 23: Centered realization spaces of polytopes, Slides