What is the name of this "generalized sum"?

Question

I'm looking for the name of this quite basic function:

$X=\sqrt[\alpha]{\sum_i{x_i^\alpha}}$ ($\alpha\in\mathbb R$)

It's a kind of "generalized sum":

• $\alpha=1$ : $X=\sum_i{x_i}$
• $\alpha\to\infty$ : $X=\max(x_i)$

I've spend hours looking on the net, but search engines are not very useful in this.

Answer 1

$$X=\left(\sum_i x_i^\alpha\right)^{1/\alpha}$$ is the $p$-norm of the vector $(x_i)_i$ where $p=\alpha$ (at least if $\alpha\ge 1$). Look here.