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I have 2 distance functions $d(x,y)=|x^2-y^2|$ and $d(x,y)=|x^3-y^3|$ and I am trying to prove that they are metrics on $\mathbb R$, or give a counterexample that they are not metrics on $\mathbb R$.

I managed to prove the first 3 properties for the 2nd function namely:

  1. $d(x,y)\ge0$

  2. $d(x,y)=0$ iff $x=y$

  3. $d(x,y)=d(y,x)$

and I now have shown that the 1st function is not a metric.

But for the triangle inequality property, $d(x,y)\le d(x,z) + d(z,y)$, Im stuck on it. How would go about proving it for $d(x,y)=|x^3-y^3|$? Thanks.

  • 1
    you made a mistake on the second property for the first of the mentioned maps, as it is not satisfied by that one.2012-02-13
  • 1
    regarding property 2: what is $d(1,-1)$ for your first function?2012-02-13
  • 0
    Thank you both, I think when I first tackled the question, I didnt think of it with negative numbers, and I was just more focused on the fourth property. Thanks again. May I ask, how would I prove the triangle inequality property holds for the 2nd function?2012-02-13
  • 1
    Hint: In the inequality you have to prove, try substituting x^3 = a, y^3 = b, z^3 = c. See anything familiar?2012-02-13
  • 0
    Thanks, is this correct? $d(x,y)=|x^3-y^3| = |a-b| \le |a-c| + |c-b| \le d(x,z) + d(z,y)$?2012-02-13
  • 0
    See also: http://math.stackexchange.com/questions/965818/which-properties-must-a-function-f-fulfill-for-d-to-be-a-metric2015-12-07

2 Answers 2