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Convex set question

  1. Jan 29, 2009 #1
    1. The problem statement, all variables and given/known data
    Let x* be an element of a convex set S. Show that x* is an extreme point of S if and only if the set S\{x*} is a convex set.

    2. Relevant equations

    (1-λ)x1 + λx2 exists in the convex set

    3. The attempt at a solution

    I'm not too sure what S\{x*}, I asssumed it was the same as S/{x*} which is S over {x*}
    I have is S?{x*} is a convex set then
    λ(K/x*) + (1-λ)(P/x*) is a convex set were K and P are in the convex set S.
    [λ(k) + (1-λ)(P)]/x* is in S/{x*}, but I can't see how it must me an extreme point
  2. jcsd
  3. Jan 29, 2009 #2


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    Homework Helper

    Usually \ is read as "minus":
    [tex]S \setminus \{ x^* \} = \{ s \in S \mid s \neq x^* \}[/tex]

    I suppose the statement is intuitive: you can only keep drawing straight lines between points, if the point you take out is on an "edge".
  4. Jan 29, 2009 #3
    Ahh thanks, I should have looked up the notation haha!
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