Comment by gus_massa
There is a local metric. The value of length/2/r depends on how big the circle is:
Imagine the Earth is a sphere. You make circles centered in the north pole:
* If the circle is tiny, the Earth is almost flat and you get almost pi.
* If the circle is the equator, you have to walk 1/4 of length the circle from the pole to the equator, so the result is 4/2=2
* If the circle is so big that you walked almost to the south pole, the result is almost 0.
That makes perfect sense!
I guess my point is that Pi is only a minimum in the selected family of metrics that the article examines. There are plenty of other metrics where Pi is as small or as big as you want.