# Plotting $l^p$ circles

I'm trying to plot a couple of $l^p$ circles, but when $p=\infty$ its not quite showing up (the max one in the code below):

ContourPlot[{Abs[x] + Abs[y] == 1, x^2 + y^2 == 1, x^4 + y^4 == 1,
Max[Abs[x], Abs[y]] == 1}, {x, -1, 1}, {y, -1, 1}]


Is there a way to plot this (preferably not telling Mathematica that it's a square)?

You need to get sampling points on both sides of the curve for it to show up with ContourPlot.

ContourPlot[{Abs[x] + Abs[y] == 1, x^2 + y^2 == 1, x^4 + y^4 == 1,
Max[Abs[x], Abs[y]] == 1}, {x, -1.05, 1.05}, {y, -1.05, 1.05}(*, Exclusions -> None*)]


Add the option Exclusions -> None to remove the gaps in the corners. Mathematica is thinking there is a discontinuity, but it's wrong.

With a little trickery, you can use Norm[v, p] and make a slider run between Infinity and 1:

Manipulate[
ContourPlot[
Norm[{x, y}, Limit[1/pp, pp -> p]], {x, -1.05, 1.05}, {y, -1.05, 1.05}],
Row[{Control[{p, 0, 1}],
InputField[Dynamic[Limit[1/pp, pp -> p], If[# >= 1, p = 1/#] &],
Appearance -> "Frameless"]}, " "]
]


RegionPlot[Evaluate[Norm[{x, y}, #] <= 1 & /@ {1, 2, 4, Infinity}], {x, -1, 1}, {y, -1, 1}]


Add the option PlotStyle -> {None, None, None, None}] to get