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.



Existence of p=-infinity norm
Yes this limit is correct (subject to Yemon Choi's comment). See for example the section on means, in the book *Analytic Inequalities* by Mitrinovic.


