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This question already has an answer here:

Does any one know of a concept analogous to $\pi$ in metric spaces. Namely, taking the all the points $1$ away from a point, and measuring the distance as some sort of limit? This was prompted when I was reading a book, I it was pondering what would happen if $\pi=3$ (although, it dealt with it by defining a circle to be a hexagon.)

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marked as duplicate by GEdgar, user61527, Hanul Jeon, Dan Rust, Shobhit Nov 11 '13 at 2:53

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  • $\begingroup$ I think you need quite a bit more structure than an arbitrary metric space to do what you want. For example, there's not really any good $\pi$ analogue on the Paris metro metric... $\endgroup$ – Micah Nov 10 '13 at 23:22
  • $\begingroup$ So it wouldn't be defined for all metric spaces. Good to know. I would also imagine it could depend upon the point chosen. $\endgroup$ – PyRulez Nov 10 '13 at 23:33
  • $\begingroup$ also math.stackexchange.com/q/53023/442 $\endgroup$ – GEdgar Nov 11 '13 at 1:34
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You could define ${\hat d}$ to be the intrinsic metric corresponding to the given metric $d$ (see http://en.wikipedia.org/wiki/Intrinsic_metric ), and define

$$p = \sup\{{\hat d}(y,z) \mid d(x,y)\leq 1, d(x,z)\leq 1,\ x,y,z\in S\}. $$

If $S$ is $\mathbb{R}^n$ and $d$ is the Euclidean metric on $\mathbb{R}^n$, then $p=\pi$. I would guess metric spaces exist for which $p=\infty$ above. If you changed the definition above so that you took an $\inf$ over $x$ rather than a $\sup$, you would still get $p=\pi$ for the case of $S=\mathbb{R}^n$ and $d$ the Euclidean metric.

EDIT: after I posted this, there were several comments that your question may be a duplicate. Please check the links given; there is probably some interesting information there.

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