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Math Help - Convex hull of the union of two convex sets

  1. #1
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    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 Rn. 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.
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  2. #2
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    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.
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  3. #3
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    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.

    Convex hull of the union of two convex sets-convexproblema.png

    Convex hull of the union of two convex sets-convexproblemb.png
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