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, J. DeLoera

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.

Suggested Schedule:

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.

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.