I have just proved $$|x+y-z|+|x-y+z|+|-x+y+z| \geq |x| + |y| + |z|$$ for all $x,y,z\in\mathbb R$ and I want now to find out when this is an equality. I just know the case $x=z=y$ but I've tried $1,2$ and $3$ for $x,y,z$ and it fits. So how do I get the other cases? And how do you prove it?
$= 2|x| + 2|y| + 2|z|$