Syllabus Detail

Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 114: Convex Geometry

Approved: 2006-02-28 (revised 2019-10-25, Kuperberg, Rademacher)
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Lecture notes written by the faculty have been used successfully in the recent past. They contain a fair number of exercises. 71 pages available for free. Alternate textbook: Convex Sets and Their Applications, by Steven Lay.
Search by ISBN on Amazon: 9780486458038
Prerequisites:
MAT 021C; (MAT 022A or MAT 067).
Suggested Schedule:

If teaching from lecture notes:

Lecture(s)

Sections

Comments/Topics

3


Fundamental definitions: Affine sets, convex set, convex hull. Examples.

3


Caratheodory’s theorem, Radon’s theorem.

3


Helly’s theorem and applications.

3


Separating and supporting hyperplanes. Faces, extreme points.

3


Sets of constant width, diameter Borsuk’s problem.

3


Polyhedra and Polytopes. Examples and main operations (e.g. Projections, Schlegel Diagrams).

3


Graphs of polytopes, Euler’s formula. Coloring problems.

3


Duality and Polarity.

3


Convex bodies and Lattices. Minkowski’s first theorem, Blichfeldt’s theorem.

If using Lay's book:

Lecture(s)

Sections

Comments/Topics

2

Definition of convex sets, convex bodies, and convex polytopes. Examples of convex polytopes and non-polytopes in dimensions 2 and 3. Examples of intuitive results and open problems, e.g., sphere packing.

20

Euclidean and convex geometry in n dimensions. k-dimensional faces of n-dimensional polytopes. Volumes of parallelepipeds and simplices. Volume of the n-sphere and multiple integrals for volumes.

Wikipedia

Definition of a regular polytope. The list of regular n-polytopes and enumeration of faces. Dihedral angles of regular polytopes and the necessary condition that the total angle of each ridge is less than 2*pi.

2

Closures, convex hulls, and Caratheodory's theorem.

3, 4

Existence of separating and supporting hyperplanes.

5Extreme points and the finite-dimensional Krein-Milman theorem.

The definition of k-extreme points. The relation between k-facets and k-extreme points. Examples of k-extreme points in non-polytopes.

14

Arithmetic with sets and Minkowski sums. Statement of the Brunn-Minkowski and Rogers-Shephard inequalities. The isoperimetic inequality as a corollary of Brunn-Minkowski.

23

Polar duals of convex sets, convex bodies, and polytopes. The correspondence between the face poset of a polytope and its dual.

Optional Advanced Topics
The classification of convex polygons that tile the plane.
The symmetrization proof of Brunn-Minkowski inequality and the isoperimetric inequality.
The topological proof of existence of regular polytopes. The definition of a Coxeter simplex.
Sphere packings and Voronoi regions. Examples of sphere packings in higher dimensions.
The largest ellipsoid in a convex body (the John ellipsoid). The definition of Banach-Mazur distance and the fact that it is bounded in each dimension.
Additional Notes:
This course should serve as a bridge between the lower division courses and more abstract upper division courses. There are a few excellent supplementary resources: Eggleston’s Convexity, Yaglom and Boltyanskii’s Convex Figures, and Ziegler’s Lectures on Polytopes. For the final part, The Geometry of Numbers by C.D. Olds, Anneli Lax, and Guiliana Davidoff is appropriate for undergraduates.