Given a vector $\mathbf {x} =(x_{1},\ldots ,x_{n})$ the p-norm is defined as $$ \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p} $$ for $p \geq 1$.

For $p=\infty$ one obtains the maximum norm $$\left\|\mathbf {x} \right\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right). $$

I was wondering what happens if we consider the case $p=-\infty$?

From some numerical experiments it seems like $$ \lim_{p\rightarrow -\infty} \bigg (\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}= \min \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right). $$ Does this result hold in general? I could not find any reference for this problem.