2
$\begingroup$

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.

  • 1
    d(a,b) iff a=b is trivial2012-10-11

1 Answers 1

2

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 $c$.
  • 0
    @Berci: Great -- the Wikipedia article even mentions travelling up hill :-)2012-10-11