For $p\geq 1,$ the $p$-norm of a vector $(x,y)\in\Bbb R^2$ is the number $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}.$
I learned this definition some time ago, but I never really understood it. Is there a useful and not too difficult to understand geometric intuition that explains it (at least for natural $p$)?