Math 341: Convex Geometry
Xi Chen
479 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA
E-mail address: [email protected]
CHAPTER 1
Basics
1. Euclidean Geometry
1.1. Vector spaces. Let R be the set of real numbers.
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Math 341 Homework 2 Solution
(1) (a) Since S = cfw_(x, y) : (x + 2/9)2 + (y 2/9)2 8/81 R2 is the disk
centered at (2/9, 2/9) with radius 2 2/9, it is convex.
(b) Since S = S1 S2 S3 S4 , where S1 = cfw_x y 0, S2 = cfw_x y 1,
S3 = cfw_x + y 0 and S4 = cfw_
Math 341 Homework 2 Solution
2.6 (p. 22) Let x1 = a1 + b1 and x2 = a2 + b2 be two points in A + B with
a1 , a2 A and b1 , b2 B. We want to show that
x1 x2 = cfw_x1 + x2 : , 0, + = 1 A + B
Since
x1 + x2 = (a1 + b1 ) + (a2 + b2 ) = (a1 + a2 ) + (b1 + b2 )
a
Math 341 Homework 1 Solution
(1) (a) Since
(A + B) + C = cfw_a + b : a A, b B + cfw_c : c C
= cfw_a + b + c : a A, b B, c C
and
A + (B + C) = cfw_a : a A + cfw_b + c : b B, c C
= cfw_a + b + c : a A, b B, c C
it follows that (A + B) + C = A + (B + C).
(b)
Math 341 Homework 3 Solution
(1) A set S is ane if x + y S for any x, y S and + = 1. Since
xy = cfw_x + y : + = 1, , 0 cfw_x + y : + = 1, xy S. So
S is convex.
(2) (a) Check the vectors x2 x1 , x3 x1 , x4 x1 are linearly independent:
1 0 0 1
1 0 0 1
0 1 0
Math 341 Homework 1 Solution
1.3 (p. 9) (a) Since
(A + B) + C = cfw_a + b : a A, b B + cfw_c : c C
= cfw_a + b + c : a A, b B, c C
and
A + (B + C) = cfw_a : a A + cfw_b + c : b B, c C
= cfw_a + b + c : a A, b B, c C
it follows that (A + B) + C = A + (B +
Math 341 Homework 4 Solution
(1) Since S is convex, Int(S) is convex. And since cfw_x1 , x2 , ., xn Int(S),
convcfw_x1 , x2 , ., xn Int(S).
(2) Since every convex set in R1 is an interval, conv(S) could be [a, b], [a, b),
(a, b] or (a, b), where a and b
Math 341 Homework 3 Solution
2.13 (p. 23) Fix a point y int(S). For a point x cl(S), relint(xy)
int(S) by 2.12. Therefore,
xy = cl(relint(xy) int(S)
So x cl(int(S) and cl(S) cl(int(S). And since cl(S) cl(int(S),
cl(S) = cl(int(S).
Consider S = cfw_2 (0,
Math 341 Homework 5 Solution
2.37 (p. 25) (a) Let B = cfw_x2 + y 2 1 be a disk and p be a point on the
circle bd(B) = cfw_x2 + y 2 = 1. Let S = B\cfw_p. First, we show that S is
convex. Let u, v S and w relint(xy). Since u, v B and B is convex,
w B. Obvio
Math 341 Homework 4 Solution
2.4 (p. 22) Note that B(x, ) = cfw_y : |y x| < and B(x, ) = cfw_z :
|z x| < .
Let y B(x, ). Then |y x| = |y x| < . So y B(x, )
and B(x, ) B(x, ).
On the other hand, let z B(x, ). Then
1
|1 z x| = |z x| <
1 z B(x, ) and z B(x