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In $\mathbb{R}^n$ consider (the norm infinity) $\|x\|=\max|x(i)|$ where $1\leq i\leq n$.

Find the precise conditions under which we have $\|x+y\|=\|x\|+\|y\|$.

Thank you for your helping. :)

Math1000
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nike
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2 Answers2

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Note that $\operatorname{arg max}(|x|)=\operatorname{arg max}(|y|)$ is a necessary condition to $\max|x_i+y_i|=\max|x_i|+\max|y_i|$. Here, $\operatorname{arg max}(|x|)= k$ such that $\max(|x|)= |x_k|$

Now,$|x_k+y_k|=|x_k|+|y_k|$ . Then either $x_k$ and $y_k$ have the same sign or at least one of then is 0. In the last case, $x=[0,\dots,0]$ or $y=[0,\dots,0]$

Andre Gomes
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  • This might be useful http://math.stackexchange.com/questions/944302/when-does-the-equality-hold-in-the-triangle-inequality – Andre Gomes Aug 22 '15 at 17:38
  • i'm not good at english language but i try to understand in your advice. :) – nike Aug 22 '15 at 18:07
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If you sum $n$ pairs of numbers, the maximum of the $n$ results is the sum of the maximums if you had to add precisely these two maximums, and if they had the same sign.

ajotatxe
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