I found this inequality but I don't know where it comes from

[Felafel]

here is the inequality:
$(\sum\limits_{i=1}^n |x_i-y_i|)^2= \ge \sum\limits_{i=1}^n(x_i-y_i)^2+2\sum\limits_{i \neq j}^n |x_i-y_i|\cdot |x_j-y_j|$
does it have a name/is the consequence of a theorem?
Thank you :)

 

[jedishrfu]

http://en.wikipedia.org/wiki/Triangle_inequality

 

[jbunniii]

I believe it is actually an equality, assuming you mean the following:
$$\begin{align}
\left(\sum_{i=1}^{n}|x_i - y_i|\right)^2
&= \sum_{i=1}^{n} \sum_{j=1}^{n} |x_i - y_i|\cdot |x_j - y_j| \\
&= \sum_{i=1}^{n} |x_i - y_i|^2 +
\sum_{i=1}^{n} \sum_{j\neq i,j=1}^{n} |x_i - y_i|\cdot |x_j - y_j|\\
&= \sum_{i=1}^{n} |x_i - y_i|^2 +
\sum_{i=1}^{n} \sum_{j=1}^{i-1} |x_i - y_i|\cdot |x_j - y_j| +
\sum_{i=1}^{n} \sum_{j=i+1}^{n} |x_i - y_i|\cdot |x_j - y_j|
\end{align}$$
By symmetry, the second two terms are equal to each other, so we have
$$\left(\sum_{i=1}^{n}|x_i - y_i|\right)^2 = \sum_{i=1}^{n} |x_i - y_i|^2 + 2\sum_{i=1}^{n} \sum_{j=1}^{i-1} |x_i - y_i|\cdot |x_j - y_j|$$
I don't know if this equality has a name, but it's a standard and often useful way to rewrite the left hand side.

 

[jedishrfu]

its the triangle inequality. see the wiki article on it.

 

[D H]

Look at jbunniii's derivation, jedishrfu. It's an identity, not an inequality.