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The reverse triangle inequality as listed on https://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality shows that

$| ||x||_2 - ||y||_2 | \leq ||x -y||_2$ for $x,y \in \mathbb R^n$, where $||x||_2 = \sqrt{\sum_{i=1}^n x_i^2}$ is the euclidean norm.

Now I wonder if the reverse triangle inequality also holds when using the square of the euclidean norm $||x||_2^2 = \sum_{i=1}^n x_i^2$ instead, i.e. does

$| ||x||_2^2 - ||y||_2^2 | \leq ||x-y||_2^2 $ hold?

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No, take $x=(1,1), y=(\frac{1}{2}, \frac{1}{2})$

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  • $\begingroup$ Thank you for the quick response! $\endgroup$ – Bazzan Jun 4 at 12:53

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