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Math Help - Convexity question

  1. #1
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    Convexity question

    Can someone help me prove this;

    I have managed to sketch them but cannot prove why its convex
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Convexity question

    Quote Originally Posted by entrepreneurforum.co.uk View Post
    but cannot prove why its convex
    Consider x=(x_i),\,y=(y_i) in S_{n-1} and t\in [0,1] then, (1-t)x+ty=((1-t)x_i+ty_i) satisfies \sum_{i=1}^n(1-t)x_i+ty_i=(1-t)\sum_{i=1}^n x_i+t\sum_{i=1}^ny_i=(1-t)+t=1 and (1-t)x_i+ty_i\geq 0 for all i=1,\ldots,n. That is, (1-t)x+ty\in S_{n-1}.
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