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Math Help - Proving a noSet is Convex

  1. #1
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    Proving a noSet is Convex

    Hi there

    How would I go about proving that "S= (x1,x2): x1 + x2 ≤ 1 , x1 ≥ 0)" is a convex set?

    Thanks

    *Sorry for the title typo
    Last edited by Lior539; May 20th 2010 at 04:44 AM. Reason: Typo
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  2. #2
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    Quote Originally Posted by Lior539 View Post
    Hi there

    How would I go about proving that "S= (x1,x2): x1 + x2 ≤ 1 , x1 ≥ 0)" is a convex set?

    Thanks

    *Sorry for the title typo

    Take any two points u:=(x_1,x_2),\,v:=(y_1,y_2)\in S ; you must prove that tu+(1-t)v\in S\,,\,\,\forall\,t\in [0,1]:

    tu+(1-t)v=\left(tx_1+(1-t)y_1,\,tx_2+(1-t)y_2\right) , and now you have the simple task yo show that:

    1) tx_1+(1-t)y_1\geq 0 ;

    2)  tx_1+(1-t)y_1+tx_2++(1-t)y_2\geq 1 ...

    Tonio
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  3. #3
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    Thanks for your help Tonio, it really cleared up some things in my mind. How would one go about proving (1) and (2) though?

    Thanks
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