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let X be the surface of the earth for any two points on the earth surface. let d(a,b) be the least time needed to travel from a to b.is this the metric on X? kindly explain each step and logic, specially for these two axioms d(a,b)=0 iff a=b and triangle inequality.

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d(a,b) iff a=b is trivial –  Bitwise Oct 11 '12 at 15:59

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This will generally not be a metric since the condition of symmetry is not fulfilled: It usually takes a different time to travel from $a$ to $b$ than to travel from $b$ to $a$. (I know that because I live on a hill. :-)

The remaining conditions are fulfilled:

  • The time required to travel from $a$ to $b$ is non-negative.
  • The time required to travel from $a$ to $b$ is zero if and only if $a=b$.
  • $d(a,c)\le d(a,b)+d(b,c)$ since you can always travel first from $a$ to $b$ and then from $b$ to $c$ in order to travel from $a$ to $c$.
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Such is called quasimetric: en.wikipedia.org/wiki/Semimetric_space#Quasimetrics –  Berci Oct 11 '12 at 16:20
    
@Berci: Great -- the Wikipedia article even mentions travelling up hill :-) –  joriki Oct 11 '12 at 16:22

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