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Math Help - Convex Hull

  1. #1
    Junior Member qspeechc's Avatar
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    Convex Hull

    Hello Everyone.

    While I was reading along in a book I came across this: It says the convex hull is
    co(\{ x_n : n\in \mathbb{N}\}) = \{ \sum _{i=1}^m \alpha _i x_i : \sum |\alpha _i|\leq 1\}
    The x_n are vectors in a vector space and the \alpha _n complex scalars. But from my understanding (and wikipedia verifies this) the definition of a convex hull requires \sum |\alpha _i|=1. Normally I'd just think this was a typo, but the text goes on to use the fact that \sum |\alpha _i|\leq 1. So are the two things equivalent or what? I can't see how they are. . Help please!
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  2. #2
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    The definitions are only equivalent, if 0 is one of the x_{n}.
    I guess one could prove the following:
    co(\{x_{n}-x_{0} : n \in \mathbb{N}\})+x_{0}={ \{ \sum _{i=0}^m \alpha _i x_i : \sum |\alpha _i|= 1\}
    Last edited by Iondor; August 30th 2010 at 06:33 AM.
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  3. #3
    Junior Member qspeechc's Avatar
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    No 0 is not one of them. I do not see how co(\{x_{n}-x_{0} : n \in \mathbb{N}\})+x_{0}={ \{ \sum _{i=0}^m \alpha _i x_i : \sum |\alpha _i|= 1\} helps...?
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  4. #4
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    Quote Originally Posted by qspeechc View Post
    No 0 is not one of them. I do not see how co(\{x_{n}-x_{0} : n \in \mathbb{N}\})+x_{0}={ \{ \sum _{i=0}^m \alpha _i x_i : \sum |\alpha _i|= 1\} helps...?
    What exactly is your problem?
    The two definitions are not equivalent.
    The exact relationship between the definitions is expressed by above equality.
    What more do you want?
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