Measuring $\pi$ with alternate distance metrics (p-norm). How/why does $\pi$ vary with different metrics in p-norms? Full question is below.

Background

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:

$$D_n\left((x_1,y_1),(x_2,y_2)\right)=(|x_2-x_1|^n+|y_2-y_2|^n)^\frac{1}{n}$$

(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:

$$y=\left(1-|x|^n\right)^\frac{1}{n}$$

I took the normal arc length formula of:

$$\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}$$

and replaced all the powers of $2$ with powers of $n$ to get:

$$\int_a^b\left(1+\left|\frac{dy}{dx}\right|^n\right)^\frac{1}{n}$$

Combining the circle with the arc length formula (and take a quarter circle and times it by 4 gave) the following integral:

$$4\int_0^1\left(1+\left|\frac{d}{dx}\left(1-x^n\right)^\frac{1}{n}\right|^n\right)^\frac{1}{n}dx$$

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.

Question

(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? 

Note

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).
Math Question: Assuming the math above is ok, is there a reason for (2,π) 
  (2,π)
   to be a minimum?


The $p=2$ is the only p-norm that has the $SO(3)$ Lie Group structure. In other words, it is rotation invariant. Try it, you can rotate the coordinate system without changing the length. You can't do that with the Taxicab metric, the length you get will change.

The deep answer to your question is that only $p=2$ has a continuous symmetry, rotation. All the other p-norms have a either a finite number of symmetries or symmetries. Here's the thing, a p-unit circle is something like every possible vector that can generated from the group glued together. Imagine starting with a small vector then copying and rotating it just a bit, attaching it to the original vector, then repeating with the copy. Eventually, you'll generate the circle. You generate any of the p-norm circles using a similar process using their symmetry, albeit a bit more opaque. So why is $p=2$ special? The circle is made with the shortest vectors, and only the shortest vectors available to construct it. Why? Because they're all the same length! This isn't true unless you're rotationally invariant and guess what? Only $p=2$ is!!


  Math/Philosophy Question: Assuming above ok, is this why we observe the metric D2
  D2
   in the real world? 


Quick run through physics. In physics, you have a theory described by a lagrangains that describes the amount of action particles do over a length of time. It's a scalar function of distance, speed, and time. The symmetries of the lagrangian correspond to conserved quantities according to Noether's Theorem. In standard classical mechanics, we assume three things,

(1) The action of a particle moving through some path along space is unchanged if you shift the start of the motion forward or backward in time. (Energy is Conserved)

(2) The action is unchanged if you shift the translate the start of the motion to another location. (Momentum is Conserved)

(3) The action is unchanged if you rotate the motion of the particle. (Angular Momentum is Conserved)

The last one isn't talked about to much, however, it's very important and highly relevant to our discussion. Since, our lagrangian, the theory for our physical theory, is invariant under rotation, there can never be a preferred direction in our theory. This means that means the metric we are using to measure distance and speed must also be rotation invariant. The $p=2$ norm is the only p-norm that'll work.
Pi is the ratio of the circumference to the diameter. Lp geometries start from a point, becomes a coss, eventually becomes a circle at p=2, becomes more square, and eventually becomes a spiky ball. These happen as we go from dimension 0 to dimension n. This thing in its n>2 is a squircle. 

The circumference gets larger and larger while the diameter does not change. This means the value of the p-norm or pi changes as n changes.  

I read "π is the Minimum Value for Pi" and a portion of the paper it referenced, "π Goes Full Circle." The diagram in minimum value paper does not reflect this evolution. The shape is not a square at infinity. The subsequent paper asserted a smooth function. That is likewise false. 

There a lot of mentions in Quora on the sphere packing problem. Spiky balls interlace so the max packing is much denser than researchers expected. 

When using these distance functions, we ignore the square roots beyond the taxicab geometry. That causes problems. It changes the continuity of the metric. Spiky balls are not continuous. 

Thanks for bringing your question back to my attention.