unit circle under different norms

Using the p-norm defined as

\[||x||_{p} = \left(\sum_{i=1}^{n} |x_{i}|^{p}\right)^{1/p}\]

for \(p>0\), we can look at what a unit circle is for each norm. Note that for \(0<p<1\) the formula above is not actually a norm as it violates the triangle inequality required to be a norm. Regardless, let's look at what a unit circle looks like in 2D for a few of the norms:

Isn't it cool how the corners of the unit circle get pushed out as the order of the norm increases!