# Convex Polytopes and Combinatorial Geometry (18.318, Topics in Combinatorics)

Instructor: Igor Pak, 2-390, pak@math.

Class Schedule: Tu,Th 11-12:30, MIT Room 4-265 (see here to find it).

## Brief outline

We will discuss a variety of combinatorial and geometric questions on convex polytopes, along with some rather diverse application. The topics will include some rather standard ones (Dehn-Sommerville equations, Helly Theorem, Cauchy and Steinitz theorems) as well as some less standard ones (Alexandrov and Minkowski existence theorems, Koebe Theorem, Sabitov's proof of the The Bellows Conjecture, nonoverlapping unfoldings). The (somewhat ambitious) content of the course is outlined below, along with the suggested reading materials.

## Course Requirements

No home assignments or exams will be given. All listeners will have to take turns in typing up lectures. There will be a research project due for those who need a grade.

## Content:

• Kissing number of a sphere is < 14 (in fact it's 12 but we won't prove that)
• Helly Theorem. Various extensions and generalizations.
• Dehn-Sommerville equations, Kalai's "simple way to tell a simple polytope"
• Perles's example of an "irrational polytope", Mnev's Universality Theorem via Lawrence construction
• Linkages, theorems of Kempe and Kapovich-Millson (without proof)
• Cauchy Theorem (geometric and algebraic proofs)
• Steinitz Theorem, rationality of 3-dim polytopes
• Sabitov polynomials for polyhedra homeomorphic to a sphere (after Connelly, Sabitov, and Walz)
• dimension of moduli spaces of polytopes, bipyramids, solution of the Robbins conjecture
• Alexandrov "existence" theorem, Volkov's approach
• nonoverlapping unfoldings, what happens in higher dimensions
• Brunn-Minkowski inequality, applications to order polytopes
• Minkowski Theorem on polytopes
• Koebe Theorem on circle packing, application to plane separator theorem
• counting integer points on polytopes, Barvinok's algorithm

P.S. I know, I know - it's a little too ambitious - the final cut will be made later...

Course Textbooks:
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.
Both are available at Quantum Books

More Textbooks:
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.

## Topic references

1. For background on kissing numbers see and Conway and Sloane, Sphere Packings, Lattices and Groups. For an interesting and (relatively) simple proof of k3 = 12 see this recent paper by Oleg Musin.

2. For Helly Theorem, various extensions and generalizations see classical survey article Danzer, Grunbaum, and Klee, Helly Theorem and its relatives (1963). In our presentation we followed chapters 8 and 9 of Matousek.

For classical introduction to Borsuk theorem (case d = 2) and Borsuk conjecture see Hadwiger and Debruner, Kombinatorische Geometrie In Der Ebene. For disproof of the conjecture see Kahn and Kalai original article. Compare with the version in Aigner and Ziegler, Proof from the Book.

3. For Dehn-Sommerville equations and Kalai's simple way to tell a simple polytope see Ziegler, sections 8.3 and 3.4. Our presentation of DS-equation followed Bronsted, An introduction to convex polytopes.

4. For Balinski theorem see Ziegler, sections 3.5. Our presentation of the Y&Delta proof of Steinitz theorem followed Ziegler's book, sections 4.1-4.3. For a classical approach and various connections see Grunbaum, chapter 13.

For Tutte's equilibrium approach see chapters 12, 13 in this book by Richter-Gebert (it was published in Springer's Lecture Notes).

For Koebe-Andreev-Thurston theorem and its applications see Pach-Agarwal book. The variational approach we presented followed section 1 in recent Ziegler's lectures notes

5. For Mnev's Universality theorem, various extensions, and references see above mentioned Richter-Gebert's book. An introduction and the Lawrence construction is given in Ziegler, section 6.6. Note also a good (and very short) discussion in Matousek, end of section 5.3.

The polytope of Micha Perles is described in Grunbaum (section 5.5) and in Ziegler (section 6.5).

6. For a background on linkages see classical books Hilbert and Cohn-Vossen, Geometry and the Imagination, chapter 5; Courant and Robbins What is Mathematics?, section 3.5.

For modern approach and Kapovich and Millson's theorem see their original article; see also King's followup.