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Math Help - A non translation/dilation-invariant metric

  1. #1
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    A non translation/dilation-invariant metric

    Hi all,

    Given a normed vector space X, the norm gives rise to a metric that satisfies
    1. d(x + z, y + z) = d(x, y) and 2. d(ax, ay) = |a|d(x, y) for all x, y, z in X.

    I'm looking for a vector space X with a metric that does not satisfy these. I've found that |x^3 - y^3| on R does not satisfy either. I can't think of any that satisfy 1 but not 2, and 2 but not 1.

    Thanks.
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  2. #2
    Senior Member roninpro's Avatar
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    Have you tried using the discrete metric?

    d(x,y)=1 if x=y
    d(x,y)=0 if x\ne y

    This should satisfy (1) but not (2).
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  3. #3
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    Quote Originally Posted by roninpro View Post
    Have you tried using the discrete metric?

    d(x,y)=1 if x=y
    d(x,y)=0 if x\ne y

    This should satisfy (1) but not (2).
    Ah - didn't think of that. But I agree, satisfies (1) but not (2). Now for the other...
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  4. #4
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    Normed vector space:
    d(x,x)=0
    d(x,y)=||x||+||y||.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by measureman View Post
    Hi all,

    Given a normed vector space X, the norm gives rise to a metric that satisfies
    1. d(x + z, y + z) = d(x, y) and 2. d(ax, ay) = |a|d(x, y) for all x, y, z in X.

    I'm looking for a vector space X with a metric that does not satisfy these. I've found that |x^3 - y^3| on R does not satisfy either. I can't think of any that satisfy 1 but not 2, and 2 but not 1.

    Thanks.
    It looks as though both your questions have been answered, but as an afterthought I'd like to add that particularly important example of this is \ell^{\infty} with the metric \displaystyle d(\{x_n\},\{y_n\})=\sum_{n=1}^{\infty}\frac{|x_n-y_n|}{2^n(1+|x_n-y_n|)} doesn't satisfy either!
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