The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$
Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need to estimate an expression of the form $|x+y+z|$ from below so that the estimate depends on the absolute values of $x,y,z$, and I think the triangle inequality should be enough, but I don't know how to do it.
Note that $$ |x+y|\ge{\large{|}}|x|-|y|{\large{|}} $$ is a combination of $$ |x|\le|y|+|x+y|\qquad\text{and}\qquad|y|\le|x|+|x+y| $$ This same idea can be applied to the standard multi-term triangle inequality $$ \left|\,\sum_{i=1}^nx_i\,\right|\le\sum_{i=1}^n|x_i| $$ To make an estimate from below of a sum, usually we have a large term and several smaller terms to get something like $$ \left|\,\sum_{i=1}^nx_i\,\right|\ge\max_j\left(|x_j|-\sum_{i\ne j}|x_i|\right) $$ Depending on the data and what you know about it, similar inequalities can be derived.
