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    Metric Space Distances and Radiuses

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    Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realise X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?

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    Solution Preview

    Metric 2
    Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realise X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?

    Firstly we will check if d is a metric.
    1. for any
    This is true, as 1 or 2.
    2. , also true since if x and y are not equal, then 1 or 2.
    3. - True
    4.
    This is true since maximum of is 2, while minimum of and .
    In this case we get . True.
    Now let T be a transformation of X into En.
    Let be the distance between points x and y of En, and let Tx, Ty be any images of x and y, respectively.
    If there is a length a >0 such ...

    Solution Summary

    Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realize X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?

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