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Math Help - Convexity of Unit ball

  1. #1
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    Convexity of Unit ball

    Am I right in saying that the unit ball is convex if for any 2 points in the unit ball, the straight line segment between them is also in the unit ball?

    The book uses Cauchy-Schwarz to prove convexity of the ball. Then they do the following:

     s^{2}|x|^2+ 2st|x||y| + t^{2}|y|^2 \leq (s+t)^2 = 1 hence  |z | \leq 1 which proves convexity of the ball.


    How did they deduce the above inequality?

     z = sx+ty where  0 \leq s,t \leq 1 .
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  2. #2
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    I am not sure of the notation because you did not tell us about x & y.
    But if they are in the unit ball: \left| x \right| < 1,\;\left| y \right| < 1,\;\left| x \right|^2  < 1,\;\left| y \right|^2  < 1.
    Make the substitutions in the inequality.

    If that is not what you need, please explain the difficulty in greater detail.
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  3. #3
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    Oh ok, I get it now.

    Thanks
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