# Discrete and Polyhedral Geometry

**Course: ** MATH 8680, Topics in Combinatorics

**Instructor: ** Igor
Pak, VinH 258, pak@math.

**Class Schedule: ** MWF 2:30, VinH 211.

## Brief outline

We will discuss a variety of combinatorial and geometric
questions on convex polytopes, along with a number of application.
The content of the course is outlined below.
The lectures will follow selected chapters from my forthcoming
book.

## Content:

- Helly theorem. Various extensions and generalizations.
- Barany theorem. The planar case via fair division.
- Dehn-Sommerville equations, Kalai's "simple way to
tell a simple polytope", Balinski theorem
- Variational approach (billiard trajectories, closed geodesics)
- Steinitz theorem, outline of two proofs
- Perles's example of an "irrational polytope",
and the Lawrence construction
- Triangulations of polygons and polyhedra, local move connectivity
- Scissor congruence in the plane, Hilbert third problem (Bricard's version
and Kagan's presentation of the Dehn invariant)
- Polytope algebra and Sydler's theorems
- Gauss-Bonnet theorem for convex polyhedra
- Cauchy Theorem (proofs by Legendre-Cauchy, Dehn, Pogorelov-Volkov)
- Examples of flexible polyhedra
- Proof of the bellows conjecture (after Connelly, Sabitov, and Walz)
- Alexandrov theorems on polytopes with vertices on rays and given curvature, Pogorelov's proof
- Alexandrov's uniqueness and existence theorems characterizing polytopes by their unfoldings
- The Brunn-Minkowski inequality and the Minkowski theorem on polytopes
- Nonoverlapping unfoldings of convex polytopes

P.S. This is a little too ambitious. Due to the time constraints I might have to cut a few
results at the end.

### Course Textbooks:

My book should
suffice. For further reading see:
J. Matousek, *Lectures on Discrete Geometry*,
Graduate Texts in Mathematics **202**, Springer, 2002.

G. Ziegler, *Lectures on Polytopes*,
Graduate Texts in Mathematics **152**, Springer, 1995.

P.M. Gruber, *Convex and discrete geometry*,
Springer, Berlin, 2007.

A. Barvinok, *Course in Convexity*,
Graduate Studies in Mathematics **54**, AMS, 2002.

B. Grunbaum, *Convex Polytopes*,
Graduate Text in Mathematics **221**, Springer, 2003.

J. Pach and P.K. Agarwal, *Combinatorial geometry*,
John Wiley, New York, 1995.

I believe all these books are available in the math library, from
Amazon.com and other retailers.

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*Last updated 8/14/2008*