Value of $\pi$ in a redifined space

Question

We know that $\pi$ is defined as the ratio of circumference and diameter of a circle. Also the circle is defined as collection of points at a fixed distance from a given point (centre). Here let us assume the centre to be origin for sake of simplicity.
We know that, in euclidean space the distance is defined as $\sqrt{x^2+y^2}$. Now if we redefine the distance as $\sqrt[n]{x^n+y^n}$ the shape of the circle (as defined above) would change and so would the value of $\pi$. Can someone derive the formula for any general $n$.
I proceeded with taking a circle of 1 unit radius. Equation of this circle would be $x^n+y^n=1$. We just have to find the perimeter of this circle and divide by $2$ to get the required $\pi$.
But the catch here is that the distance is redefined. So, if we try to use integration to calculate perimeter by taking small elements, the length of this small elements would be $\sqrt[n]{x^n+y^n}$ and not $\sqrt{x^2+y^2}$.
Due to this I get a complex integral which I am unable to solve analytically. Is there any other way round?

Answer 1

The definition of $\pi$ is not

"the ratio of circumference and diameter of a circle"

One of its possible definitions is

the ratio of circumference and diameter of a circle in the Euclidean geometry.

In hyperbolic geometry (as an example) the circumference of a circle of radius $r$ is

$$2\pi \sinh(r).$$

(Here the parameter of the hyperbolic plane is $1$.)

So the "hyperbolic $\pi$" is not

$$\frac{2\pi \sinh(r)}{2r}=\frac{\pi\sinh(r)}r.$$

Such a "$\pi$" would not even be a constant. It is an Euclidean specialty that the circles are similar and, as a result, we can define $\pi$ as constant. The real surprise is that this rudimentarily Euclidean constant appears so may times without Euclidean geometry.