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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.

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2 Answers 2

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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.

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Thank you very much! I understand. –  Bartek Feb 3 '13 at 21:08

i think yes,just look here http://en.wikipedia.org/wiki/Triangle_inequality

hopefully it can help you :)

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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

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