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Topological space metrics

  1. Aug 5, 2008 #1
    A metric dp on the topological space X×Y, with dX(x,y) and dY(x1,y1) being metrics on X and Y respectively, is defined as

    dp((x,y),(x1,y1))=((dX(x,y))p+(dY(x1,y1))p)1/p

    What does each dp((x,y),(x1,y1)) mean (geometrically or visually)?

    as p[tex]\rightarrow[/tex][tex]\infty[/tex],

    d[tex]\infty[/tex]=max((dX(x,y),dY(x1,y1)).

    What is the meaning of "max(A,B)"?

    Each of the dp((x,y),(x1,y1)) are strongly equivalent; what does this mean geometically?
     
  2. jcsd
  3. Aug 6, 2008 #2

    morphism

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    Re: Metrics

    You probably want p>=1 for d_p to actually be a metric. Anyway, have you tried putting X=Y=R? This should be revealing, e.g. if p=1 you get the "taxicab metric" and if p=2 you get the usual Euclidean metric. For other p>1, you may want to sketch the unit balls of (R^2, d_p) to get a feel for what the metric d_p does.

    max(A,B) means what you would expect it to be; namely, max(A,B) is A if A>=B and B otherwise.

    Finally, two metrics d and d* on a metric space Z are said to be strongly equivalent if you can find positive constants M and m such that the string of inequalities
    m d(x,y) < d*(x,y) < M d(x,y)
    holds for all x and y in Z. This essentially means that inside each d-ball you can find a d*-ball and vice versa. In particular, this implies that the metrics d and d* generate the same topology on Z.
     
    Last edited: Aug 6, 2008
  4. Aug 6, 2008 #3
    Re: Metrics

    Thanks. I've been worrying about what they were fr a while. It seems kinda obvious now.

    Thanks again!
     
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