Convex hull of the union of two convex sets

Hey, this is my first post so if this is posted in the wrong place just tell me.

Let A and B be convex sets in R^{n}. Show that [A ∪ B] = {sa + tb: a ∈ A, b ∈ B, s,t ∈ R, s + t = 1}

I already showed that {sa + tb: a ∈ A, b ∈ B, s,t ∈ R, s + t = 1} ⊆ [A ∪ B]. Also clearly A ∪ B ⊆{sa + tb: a ∈ A, b ∈ B, s,t ∈ R, s + t = 1}. So I just need to show that the given set is convex, which is what I'm having trouble with.

Re: Convex hull of the union of two convex sets

Hey Nezi.

Hint: Something is convex if for points A and B inside the set that ta + (1-t)b is also in the set. if s + t = 1 then s = 1 - t.

2 Attachment(s)

Re: Convex hull of the union of two convex sets

Hi,

I've attached a solution. The solution given is "brute force"; there probably is an easier, more elegant solution.

Attachment 28444

Attachment 28445