I have the following statement in my notes:
"You might want to check by drawing the sets of all $x\in\mathbb R^2$ such that $\|x\|_1=1$,$\|x\|_2=1$,$\|x\|_\infty=1$ that indeed these norms are equivalent."
Let $x:=(x,y)$. I have managed to sketch the sets using wolfram alpha, with each drawing corresponding the respective norm as follows:
My question now is how is it exactly that these drawings show that the three norms are equivalent? What can we take from them shows this fact? Has it to do with the fact that they are all within $|x|,|y|=1$?