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Math Help - convex hull

  1. #1
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    Jul 2011
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    convex hull

    Hi,

    I got the following question in a research project.

    Consider n vectors x1,...,xn in R^m satisfying the following property (*):

    (*) no vector is coordinate-wise smaller than any convex combination of the other n-1 vectors.

    I want to show that for any epsilon>0, I can find another n vectors y1,...,yn such that

    (1) y1 is in the epsilon-ball of x1, y2 is in the epsilon-ball of x2,......,yn is in the epsilon-ball of xn;
    (2) y1,...,yn still satisfy (*)
    (3) each vector of y1,...,yn is outside the convex hull of the other (n-1) vectors.

    I believe it's true and I also see how it works when m=2.

    I greatly appreciate your help.
    Last edited by vivian6606; January 15th 2012 at 10:54 PM.
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