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Thread: Problem with inequality

  1. #1
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    Problem with inequality

    Hi!, I have problems with this Proposition

    PROPOSITION

    Let d,d^{\prime} and d^{\prime\prime} metric space. For any x,y \in{\mathbb{R^n}} we have

    d(x,y)\leq{d^{\prime}(x,y)}\leq{nd^{\prime\prime}(  x,y)} , where d(x,y)= [ \displaystyle\sum_{i=1}^N (x_i-y_i)^2]^{1/2} ,

    d^{\prime}(x,y)= \displaystyle\sum_{i=1}^N |x_i-y_i| , max_{1\leq{i}\leq{n}}  |x_i-y_i|

    I have problems with proof the inequalities
    Last edited by cristianoceli; Jan 17th 2018 at 08:22 AM.
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  2. #2
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    Re: Problem with inequality

    Quote Originally Posted by cristianoceli View Post
    Hi!, I have problems with this Proposition

    PROPOSITION

    Let $d,d^{\prime}$ and $d^{\prime\prime}$ metric space. For any $x,y \in{\mathbb{R^n}}$ we have

    $d(x,y)\leq{d^{\prime}(x,y)}\leq{nd^{\prime\prime} (x,y)}$ , where $d(x,y)= [ \displaystyle\sum_{i=1}^N (x_i-y_i)^2]^{1/2}$ ,

    $d^{\prime}(x,y)= \displaystyle\sum_{i=1}^N |x_i-y_i|$ , $max_{1\leq{i}\leq{n}} |x_i-y_i|$

    I have problems with proof the inequalities
    I have replaced "tex" and "/tex" with dollar signs.
    Thanks from cristianoceli
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  3. #3
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    Re: Problem with inequality

    It is a geometric property of triangles that the sum of the lengths of two sides is always greater than the third. So, for 2 dimensions, the inequality follows immediately from the fact that d(x,y) is the length of the hypotenuse while d'(x,y) is either that exact same length or it is the sum of the lengths of the other two sides of the triangle. For higher dimensions, you would need a generalization of that fact.

    http://www.cs.bc.edu/~alvarez/NDPyt.pdf

    This gives an idea of how to start.
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  4. #4
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    Re: Problem with inequality

    Thanks!!!!
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