Unit circle under different norms

There are many ways to calculate distance in $R^n$, one family of them are called p-norms. (norm is the distance to the origin.)

\begin{align*} D &= (\sum |x_i|^p)^{1/p} \end{align*}

Among them 1-norm is called the Taxicab-norm, 2-norm the Euclidean norm. Below is a plot of unit circles under different norms. Note how the curvature changes with p!

Unit circle under different norms (mathematica code)

Click to toggle code (may have to try twice)

ContourPlot[{
Abs[x]^(1/2)+Abs[y]^(1/2)==1,
Abs[x] + Abs[y] == 1,
x^2+y^2==1,
Abs[x]^3+Abs[y]^3==1,
Max[Abs[x], Abs[y]] == 1
},
{x, -1.1, 1.1}, {y, -1.1, 1.1},
ContourLabels -> All,
PlotLegends -> "Expressions",
ContourStyle -> {Hue[1/5], Hue[2/5], Hue[3/5], Hue[4/5], Hue[5/5]}
]