Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer
Thank you very much! I understand. – Bartek Feb 3 '13 at 21:08

i think yes,just look here

hopefully it can help you :)

share|cite|improve this answer
Could you point to the particular part of the article where this is explained? I don't see it there. – Bartek Feb 3 '13 at 20:15
Please supply more than just a link. Links can go stale and then the answer loses its value. – robjohn Feb 3 '13 at 20:23

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.