Convex Geometry
Math 92.490/651 ("Selected Topics")
Spring 2010
Instructor: D. Klain
Lectures: M 3:00-5:50pm in Olsen 406
Office Hours: MWF 2pm and by appointment
Office: OH 428N
Text: Part 1 of the textbook can be downloaded here: pdf
(stay tuned for updates...)
This course provides an introduction to the geometry of convex sets in Euclidean space.
Topics will include the convex sets and convex functions, Jensen's inequality and its
relatives,
support and separation, Minkowski sums, polygons and polyhedra,
volume, surface area, mean width and related functionals, isoperimetry,
containment, geometric probability,
and inverse questions regarding shadows and cross-sections of convex bodies
(integral geometry and geometric tomography).
Prerequisites are Calculus III and some linear algebra. Some real analysis background
is helpful, but not required. The course is co-listed for both undergraduate
and graduate students.
There will be no exams.
Grades will be based on homeworks, collected in class each week.
Homework assignments will be announced weekly on this page. Many of the exercises in embedded in the text.
Check this site periodically for updates.
Readings: Sections 1, 2
Problem Set:
Section 1, Exercises 1.2, 1.4, 1.5
Section 2, Exercises 2.1, 2.4, 2.5, 2.6, 2.7, 2.10
due Monday, February 1.
Readings: Sections 2, 3
Problem Set:
Section 2, Exercises 2.9, 2.12, 2.13
Section 3, Exercises 3.3, 3.4, 3.5, 3.6
due Monday, February 8.
Readings: Sections 3, 4
Problem Set:
Section 3, Exercises 3.11, 3.13
Section 4, Exercises 4.2, 4.3, 4.4, 4.5, 4.6, 4.7
due Tuesday, February 16. (Monday is a holiday this week.)
Readings: Sections 4, 5
Problem Set:
Section 4, Exercises 4.19, 4.20, 4.21, 4.23, 4.24, 4.25, 4.28, 4.29
due Monday, February 22.
Note: The course notes have been updated. See link at the top, or click
here: pdf
Readings: Section 6
Problem Set:
Section 5, Exercise 5.1
Section 6, Exercises 6.5, 6.8, 6.9, 6.18, 6.19, 6.20
Note: There is a typo in 6.19. The "RR" should be "R" (the set of all real numbers).
due Monday, March 1.
Readings: Sections 6 and 12
Problem Set:
Section 6, Exercise 6.17
Section 12, Exercises 12.1, 12.2, 12.4, 12.6, 12.14, 12.17
due Monday, March 8.
Readings: Sections 12 and 13
Problem Set:
Section 12, Exercises 12.9, 12.10, 12.13, 12.16
Section 13, Exercises 13.3, 13.7, 13.8, 13.9, 13.13
due Monday, March 22.
Note: There are no classes on March 15 (Spring Break).
Readings: Sections 14 and 15
Problem Set:
Section 14, Exercises 14.1
Section 15, Exercises 15.2, 15.3, 15.5, 15.6, 15.7, 15.16
For problems in Section 15, recall that any bounded sequence of real numbers
contains a subsequence that converges to a limit. Similarly, any bounded sequence
of points in finite-dimensional real space contains a convergent subsequence that
converges to a limit point.
due Monday, March 29.
Readings: Sections 15, 16, 18
Problem Set:
Section 15, Exercises 15.12, 15.13, 15.14, 15.15, 15.17
Section 18, Exercises 18.3, 18.5, 18.12
For problems in Section 15, note that we proved Exercise 15.11 in class.
due Monday, April 5.
Note: The course notes have been updated. See link at the top, or click
here: pdf
Readings: Section 18
Problem Set:
Section 18, Exercises 18.10, 18.11, 18.13, 18.16, 18.17, 18.18, 18.19, 18.20
due Monday, April 12.
Readings: Section 20, and extra notes.
Problem Set:
Section 20, Exercises 20.8, 20.9
Supplementary exercises: pdf
due Monday, April 26.
Here are some extra course notes on volumes and determinants: pdf
There are no classes on Monday, April 19.
Readings: Section 21, and class hand-out on mixed area.
Problem Set:
Section 23 (in handout), Exercises 23.3, 23.5
Supplementary exercises: pdf
due Monday, May 3.
Readings: Class hand-out on mixed area.
Problem Set:
Supplementary exercises: pdf
due Monday, May 10.
Readings: Sections 25, 26 in updated notes updated again on 5/11/2010
Problem Set:
Section 25 (in updated notes), Exercises 25.8, 25.10, 25.11, 25.12, 25.16
due Monday, May 17 by 3pm in my office (Olney 428N).
My homepage (including contact information) can be found here.