Convex Polytopes and Combinatorial Geometry
(18.318, Topics in Combinatorics)
Pak, 2-390, pak@math.
Class Schedule: Tu,Th 11-12:30, MIT Room 4-265
(see here to find it).
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
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.
- 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
- 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
- Brunn-Minkowski inequality, applications to
- Minkowski Theorem on polytopes
- Koebe Theorem on circle packing, application
to plane separator theorem
- counting integer points on polytopes,
P.S. I know, I know - it's a little too ambitious - the final
cut will be made later...
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
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.
- 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
recent paper by Oleg Musin.
- 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
article. Compare with the version in
Aigner and Ziegler, Proof from the Book.
- 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.
- 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,
For Tutte's equilibrium approach see chapters 12, 13 in
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
- For Mnev's Universality theorem, various extensions, and references
see above mentioned Richter-Gebert's
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).
- 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
article; see also King's
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Last updated 1/27/2005