Whoever finds a norm for which $\pi=42$ is crowned nerd of the day!

Can the principle of $\pi$ in euclidean space be generalized to 2-dimensional metric/normed spaces in a reasonable way?

For Example, let $(X,||.||)$ be a 2-dimensional normed vector space with a induced metric $d(x,y):=\|x-y\|$. Define the unit circle as $$\mathbb{S}^1 := \{x\in X|\;\|x\|=1\}$$ And define the outer diameter of a set $A\in X$ as $$d(A):=\sup_{x,y\in A}\{d(x,y)\}=\sup_{x,y\in A}\{\|x-y\|\}$$ Now choose a continuous Path $\gamma:[0,1]\rightarrow X$ for which the image $\gamma([0,1])=\mathbb{S}^1$. Using the standard definition of the length of a continuous (not necessarily rectificable) path given by $$ L(\gamma):=\sup\bigg\{\sum_{i=1}^nd(\gamma(t_i),\gamma(t_{i+1}))|n\in\mathbb{N},0\le t_0\lt t_1\lt ... \lt t_n\le 1\bigg\}$$ we can finally define $\pi$ in $(X;\|.\|)$ by $$\pi_{(X,\|.\|)}:=\frac{L(\gamma)}{d(\mathbb{S}^1)}$$ (This is way more well-defined than the old definition below)

Examples:

• For the euclidean $\mathbb{R}^2$, $\pi_{\mathbb{R}^2}=3.141592...$
• For taxicab/infinity norms, $\pi_{(\mathbb{R}^2,\|.\|_1)}=\pi_{(\mathbb{R}^2,\|.\|_\infty)}=4$
• For a norm that has a n-gon as a unit circle, we have $\pi_{(\mathbb{R}^2,\|.\|)}=??$ (TODO: calculate)

While trying to calculate values for $\pi$ for interesting unit circles, I have defined a norm induced by a unit circle: let $\emptyset\neq D\subset X$ be star-shaped around $0\in D$. Define $\lambda D:=\{\lambda d|d\in D\}$. Now the norm in $X$ is defined as $\|x\|:=\lambda:x\in\partial(\lambda D)$ (thanks to the star-shaped-ness of $D$, this is unique.)

In other words: the scaling factor required to make x a part of the border of D. This allows us to easily find norms for most geometric shapes, that have exactly that Geometric shape as a unit circle and have the property, that the choice of the radius for the definition of $\pi$ is insignificant. (for example $\pi=6$ is calculated for regular triangles with (0,0) in the centroid)

Questions Any other interesting norms? Is this definition reasonable, and is there any practical use to this? Feel free to share your thoughts. Mind me if I made some formal mistakes. And especially, how do I define a norm with $\pi=42$?