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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 $p$)?

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    I recently found a nice article, about [$π_p$, the value of $π$ in $ℓ_p$](http://math.stanford.edu/~vakil/files/07-0333.pdf)2012-07-16

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