Lower Bound for the square root of the sum of squares

Question

I am looking for a good lower bound for the square root of the sum of squares:

Let's say we have some known parameters : $x_i > 0$ where i $\in [1,...,n]$

I am looking for a good lower bound of this term (a sum or something similar but not a one square root term):

$\sqrt{\sum_{i}x_i^2}$

PS: $\min{x_i}$ is a lower bound but not of a good quality.

Thank you.

Answer 1

The bound $\sqrt{n} \min x_i$ is better than $\min x_i$, and is sharp, in the sense that it becomes an equality, in case the $x_i$ are all equal.

EDIT: Sorry - I just saw the comment on your answer. The one in the link is better than mine, in that it is generally greater than the one I gave.

SECOND EDIT: That is, $(\sum x_i)/\sqrt{n}$.