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Uniformly equivalent metrics

  1. Jul 5, 2010 #1


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    Well, again I'm a bit stuck.

    I have to prove that the metrics d1, dp (where p is from <1, ∞>) and d∞ in R^n are uniformly equivalent. The metrics are given with:

    d1(a, b) = ∑|ai - bi|
    dp(a, b) = (∑|ai - bi|^p)^(1/p)
    d∞(a, b) = max{|ai - bi|, i = 1, ... ,n} (of course, the sums are ranging from 1 to n)

    The relation of uniform equivalence between metrics is an equivalence relation, so if d1 ~ d∞ and d1 ~ dp, then dp ~ d∞.

    I have shown that d1 ~ d∞, but I am stuck with showing that d1 ~ dp, and would be most grateful for a push here.
  2. jcsd
  3. Jul 5, 2010 #2


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    Perhaps an idea would be to use the fact to compare the p-th power of ∑|ai - bi| with ∑|ai - bi|^p ? I'm not sure if in general (∑|ai - bi|)^p (i.e. a multinomial expansion) is greater than ∑|ai - bi|^p ?
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