1. Consider the two points x 1 = (1, 1), x 2 = (1, 0) in R2 . Draw the set of points which can be obtained by using the following operations.

(a) x = 1x 1 + 2x 2 , 1, 2 R (linear combination)

(b) x = 1x 1 + 2x 2 , 1 + 2 = 1 (affine combination)

(c) x = 1x 1 + 2x 2 , 1, 2 0 (nonnegative linear combination)

(d) x = 1x 1 + 2x 2 , 1 + 2 = 1, 1, 2 0 (convex combination)

2. Let Ci , i = 1, ..., m be convex sets. Show that Tm i=1 Ci is convex.

3. Let C, D Rn be convex sets. Define the set C + D = {x + y : x C, y D}. Show that C + D is convex.

4. Let a Rn and b R. Show that the set {x Rn : a 0x b} is convex.

5. Let A be a matrix of size m n and b Rm. Show that the set P = {x Rn : Ax b} is convex. (Note that P is the set of feasible solutions to a linear programming problem.) 6. Suppose that f1, f2 are convex functions from Rn into R and let f(x) = f1(x) + f2(x). Show that f is a convex function.

7. Suppose that f1, ..., fm are convex functions from Rn into R and let f(x) = Pm i=1 fi(x). Show that f is a convex function. 8. Let f : Rn R be a convex function. Prove that the function g(x) = f(x), 0, R is a convex function.

1. Consider the two points r = (-1, 1), r? = (1,0) in R . Draw the set of points which can be obtained by using the following operations. (a) r= Mir + Azr', M1, A2 ER (linear combination) (b) x = Miel + Azr?, Al + A2 = 1 (affine combination) (c) r= Mel + Azr?, Al, A2 2 0 ( nonnegative linear combination) (d) r= Mir + Azr', Al + A2 = 1, Al, A2 2 0 (convex combination) 2. Let Ci, i = 1,..., m be convex sets. Show that ( C; is convex. 3. Let C. D C R" be convex sets. Define the set C + D = {ity : re C, y ( D}. Show that C + D is convex. 4. Let a e R" and be R. Show that the set {r e R" : a'x