Ee364a Homework 1 Solutions

θ 1 + · · · + θ k = 1. Show that θ 1 x 1 + · · · + θ k x k ∈ C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x 1 , x 2 , x 3 ∈ C, and θ 1 + θ 2 + θ 3 = 1 with θ 1 , θ 2 , θ 3 ≥ 0. We will show that y = θ 1 x 1 + θ 2 x 2 + θ 3 x 3 ∈ C. At least one of the θ i is not equal to one; without loss of generality we can assume that θ 1 = 1. Then we can write y = θ 1 x 1 + (1 − θ 1)(µ 2 x 2 + µ 3 x 3) where µ 2 = θ 2 /(1 − θ 1) and µ 2 = θ 3 /(1 − θ 1). Note that µ 2 , µ 3 ≥ 0 and µ 1 + µ 2 = θ 2 + θ 3 1 − θ 1 = 1 − θ 1 1 − θ 1 = 1. Since C is convex and x 2 , x 3 ∈ C, we conclude that µ 2 x 2 + µ 3 x 3 ∈ C. Since this point and x 1 are in C, y ∈ C. 2.2 Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine. Solution. We prove the first part. The intersection of two convex sets is convex. Therefore if S is a convex set, the intersection of S with a line is convex. Conversely, suppose the intersection of S with any line is convex. Take any two distinct points x 1 and x 2 ∈ S. The intersection of S with the line through x 1 and x 2 is convex. Therefore convex combinations of x 1 and x 2 belong to the intersection, hence also to S.