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Math Help - Minkowski distance

  1. #1
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    Minkowski distance

    Hi,
    i'm looking for the proof of minkowski metric in infinite that become the largest of the differences of the coordinates of x and y.
    can anybody help me please?
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  2. #2
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    I cannot understand anything you said?
    ...Infinite?
    ...Minkowski metric? there is no such thing?

    This is my 17th Post!!!
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  3. #3
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    Quote Originally Posted by wosci
    Hi,
    i'm looking for the proof of minkowski metric in infinite that become the largest of the differences of the coordinates of x and y.
    can anybody help me please?
    I really don't understand what you are asking for either. However, for the record the Minkowski metric is either
    \begin{bmatrix} 1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1 \end{bmatrix}

    or

    \begin{bmatrix} -1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1 \end{bmatrix}

    depending on your choice of style. (The upper left corner is the g_{00} entry.)

    -Dan
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  4. #4
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    hello,

    lim d(a,b) = Dmax(a,b)
    when r-> infinite

    d(a,b) is minkowski distance
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  5. #5
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    Quote Originally Posted by wosci
    hello,

    lim d(a,b) = Dmax(a,b)
    when r-> infinite

    d(a,b) is minkowski distance
    The Minkowski distance between two points x,\ y in \mathbb{R}^N of order p is:

    <br />
d_p(x,y)=\left[  \sum_1^N  |x_i-y_i|^p  \right]^{1/p}<br />

    The problem here is to show that:

    <br />
\lim_{p \to \infty} d_p(x,y)= \max_{i=1,..N} |x_i-y_i|<br />

    To prove this we observe that if we let z_i=|x_i-y_i| then:

    <br />
d_p(x,y)=D(z,p)=\left[  \sum_1^N  z_i^p  \right]^{1/p}<br />

    and if z_j=\max_{i=1..N} z_i

    <br />
z_j \le D(z,p) = z_j \left[1+\sum_{i=1,..N;\ i \ne j} \left(\frac{z_i}{z_j}\right)^p \right]^{1/p} \le z_j\ \left[1+(N-1) \right]^{1/p}<br />

    that is:

    <br />
z_j \le D(z,p) \le z_j\ \left[N \right]^{1/p}<br />

    But the limits of both ends of this chain of inequalities as p \to \infty are the same and equal to z_j, hence:

    <br />
\lim_{p \to \infty} D(z,p)=z_j=\max_{i=1..N} z_i <br />
,

    which proves the required result.


    RonL
    Last edited by CaptainBlack; July 17th 2006 at 10:50 PM.
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  6. #6
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    Thanks alot
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by wosci
    Thanks alot
    That's OK - note I have left some of the minor detail out, which you may
    want to fill in yourself.

    RonL
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