Today I was taking a look at Youtube video suggestions when the above equation appeared in a thumbnail. Instead of watching the video, I started to think about it a little bit. (I beg you pardon if the question ends up sounding excessively silly.)
The first thing that came to my mind was hyperbolic geometry, where indeed it's possible to make sense of such an equation: just think of a circle of radius $r=1$ around the central point of a horse cell: taking its perimetre to be equal to $2\pi r$, then we'll certainly get $\pi>3.1415...$, in particular it might very well be equal to $4$, depending on the geometry of our horse's butt. (Note however that the hyperbolic plane is such that EVERY point is a cell point, and not just a central one.)
My next thought was about the maximum norm on the Cartesian plane. Here once again, drawing a circle of radius $r=1$ around the origin, we'll get the perimetre to be exactly $8$, which would mean $\pi=4$. Of course, drawing a circle around any point in the plane would yield the same result. Nevertheless, let's forget for a moment about the maximum norm, which is somewhat ill-behaved (for instance, it gives rise to infinitely many 'geodesics' joining any two points in the plane...) and consider the $p$-norm instead, where $3\le p<\infty$, which also gives $\pi>3.1415...$. Finally, glueing these two thoughts, I ask:
QUESTION
Does the $p$-norm, with $3\le p<\infty$, induce some structure in the Cartesian plane making it, in some sense, 'isomorphic' to the hyperbolic plane?
I would say no. The notion of isomorphic you might want is called isometry. The hyperbolic plane is not isometric to $\mathbb{R}^2$ with any norm.
One important difference is that in $\mathbb{R}^2$ the diameter of the unit circle is proportional to its radius, i.e. the perimiter of a circle of radius $r$ is $p(r) = 2\pi r$ for any $r$.
In the hyperbolic plane (of constant curvature $-1$) however we have a diameter of $p(r) = 2\pi \sinh (r)$ and $\sinh (r)$ grows exponentially depending on $r$.
Another way in which they are different is that circles in the hyperbolic plane are round, you can rotate them by any angle. There are infinitely many ways in which you can rotate circles, technical statement: The isometry group of a hyperbolic circle is infinite.
For $p > 2$ the unit circle can only be rotated in four ways. It has the symmetry group of a square.