Math 114: Convex Geometry Homework 2
This problem set is due Friday, January 20. In class I have used the notation pq for the segment from p to q. But since the same bar notation is also used for closures of sets, in this problem set I will use I(p, q) fo
Math 114: Convex Geometry Homework 1
This problem set is due Friday, January 13. If a problem is starred, that means that it could be harder and it will be graded as extra credit. A particular starred problem might be within reach for you, or it might be
Math 114: Convex Geometry Solutions to the Second Midterm
1. Find the volume of the simplex in R4 with vertices at (0, 0, 0, 0), (0, 0, 0, 2), (0, 0, 1, 2), (0, 1, 1, 2), and (1, 1, 1, 2). Solution: From Yuya Kono:
1
2. Let K R2 be the hexagon with vertic
Math 114: Convex Geometry Solutions to the First Midterm
1. There exists a polyhedron K in R3 whose facets are triangles and regular pentagons, and such that each edge is shared by a triangle and a pentagon. Also 12 of the facets of K are pentagons. How m
Math 114: Convex Geometry Homework 9
This problem set is due Monday, March 19. The starred problems are due by midnight on the day of the final. (Or, if you slip them under my door, I don't expect to be in before 8am on Tuesday.) One of the starred proble
Math 114: Convex Geometry Homework 8
This problem set is due Friday, March 9.
GK8.1. For each of these non-polytopal convex bodies, identify the extremal points, and more generally the generalized k-faces in the sense of k-extremal points. (a) The interse
Math 114: Convex Geometry
Homework 7
This problem set is due Friday, March 2.
GK7.1. Recall, as I explained in class: If Bn Rn is the round unit ball, and if A is a matrix,
then A(Bn ) is an ellipsoid whose axes are the eigenlines of AAT and whose semirad
Math 114: Convex Geometry
Homework 6
This problem set is due Friday, February 24.
GK6.1. In class I introduced the idea of arithmetic with sets. Namely, that A + B is the set of all
a + b with a A and b B, that rA is the set of all ra with a A, etc. In th
Math 114: Convex Geometry
Homework 5
GK5.1. Let K be a k-dimensional parallelipiped in Rn , and let M be a matrix whose columns are
vectors of the edges of K that meet one of its corners. In class I proved the formula
Vol K = | det M |
(1)
when n = k. How
Math 114: Convex Geometry
Homework 4
This problem set is due Friday, February 10.
Let =
5+1
2
be the golden ratio.
GK4.1. Recall from lecture that you can obtain a regular dodecahedron as the convex hull of the points
(1, 1, 1), (0, 1/ , ), (1/ , , 0), an