Is there a lower-bound version of the triangle inequality for more than two terms?
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.