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Math Help - Question on the infinity metric

  1. #1
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    Question on the infinity metric

    Hi guys,

    We were learning about metric functions. I don't understand why d-infinity(0,1) is a square.

    Similarly, why is a unit ball in R3 using the infinity metric a cube?

    Can someone please explain this to me? Thanks very much.
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  2. #2
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    I am afraid you will have to explain exactly what "d-infinity" you mean. The only "infinity metrics" I know are on function spaces ( L_\infty) and sequence spaces ( l_\infty). I don't know of any "d-infinity" on Euclidean spaces.
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  3. #3
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    I am not sure if I am posting in the right section, but here it goes.

    d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)

    It is also called the supremum metric.

    Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why.

    Definition and examples of metric spaces

    In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.

    Can someone please explain why that is so?

    Thanks very much.
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  4. #4
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    Quote Originally Posted by sakodo View Post
    I am not sure if I am posting in the right section, but here it goes. d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)
    It is also called the supremum metric.
    Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why. Definition and examples of metric spaces
    In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.
    First, the link does help. Now this completely standard material.
    Second, What you first posted is misleading.
    The set \left\{ {x \in R^3 :d_\infty  (0,x) \le 1} \right\} is a cube with center at the origin.
    Here are some points in that set (1,0,0),~(0.5,-0.6,0),(1,-1,1).
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  5. #5
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    Quote Originally Posted by sakodo View Post
    I am not sure if I am posting in the right section, but here it goes.

    d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)

    It is also called the supremum metric.

    Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why.

    Definition and examples of metric spaces

    In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.

    Can someone please explain why that is so?

    Thanks very much.
    Just a side note, sakodo: if you're unsure of where a thread belongs, report your own post (lower left corner, looks like a triangle with an exclamation mark inside), and a moderator will move it for you. I would recommend that this thread go to the Analysis subforum.
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  6. #6
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    Quote Originally Posted by sakodo View Post
    d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)

    It is also called the supremum metric.

    Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why.

    Definition and examples of metric spaces

    In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.
    It doesn't actually say "d-infinity(0,1) in R2 is a square". It says that the unit circle \{x\in\mathbb{R}^2:d(x,0)=1\} is a square, for that metric d.

    The unit circle is the set of points whose distance from the origin (as measured by the d-metric) is 1. If x = (x_1,x_2) then d(x,0) = \max\{|x_1|,|x_2|\}. The equation d(x,0)=1 says that either |x_1| = 1 and |x_2|\leqslant1, or |x_1| \leqslant 1 and |x_2| =1. If you think about what that means, it says that the point x lies on the square with vertices at (\pm1,\pm1).
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  7. #7
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    Thanks for the reply guys. I think I kind of get it now.

    So in R2, the unit circle of the d-metric is a set of points of which the maximum components are less than 1. This means the bounds for this metric are the lines x=1,-1 and y=1,-1. Thus the unit circle is a square. Is this correct? Same argument goes for R3.

    But the unit ball B(0,1) , using the infinity metric, would be a square. I think I mixed the two up.

    Thanks a lot guys.
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