How/why does $\pi$ vary with different metrics in p-norms? Full question is below.
Long ago I did an investigation on Taxicab Geometry using basic geometry. One think I recall is that a circle (as defined by all points equal distance from a centre point) 'looks' like a diamond. The 'circumference' of this circle is 8. As an extension I looked at other metrics of the form:
(My limited reading of wikipedia suggests I should call this a p-norm.)
More recently using differing values of $n$ I calculated the 'circumference' of unit circles in these metrics. I took the definition of a unit circle to be all points a distance of one unit from the origin. This gave me a formula for a semi-circle:
I took the normal arc length formula of:
and replaced all the powers of $2$ with powers of $n$ to get:
Combining the circle with the arc length formula (and take a quarter circle and times it by 4 gave) the following integral:
Then $\pi(n)$ is found by dividing 'circumference' by two (twice the radius). Doing so led to this graph of $\pi(n)$ against $n$.
Interestingly $n=2$ is a minimum (both local and absolute) making our commonly thought of value of $\pi$ special.
(EDIT) Math Question: Is my distance formula for a different metric correct? (Moishe Cohen's comment suggests it might not be).
Math Question: Assuming the math above is ok, is there a reason for $(2,\pi)$ to be a minimum?
Math/Philosophy Question: Assuming above ok, is this why we observe the metric $D_2$ in the real world?
I have not formally studied metrics, tensors or vector spaces or related topics (but am happy to do some light reading if your answer requires it).