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I want an inequality of the form : $\Vert a - b \Vert^2 \leq k.(\Vert a\Vert^2 + \Vert b\Vert^2)$ ? where k is a constant.

The norm in consideration is the euclidean norm, and $a$ and $b$ are vectors in $\mathbb{R} ^p$.

As a few people have replied below, its pretty straightforward with k = 2. But I was wondering if there is something tighter than that? Clearly k = 1 if a and b are independent (if random) or orthogonal.

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    This is wrong for any norm: try $b=-a\ne0$.2012-06-09
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    Maybe you want $\Vert a - b \Vert^2 \leq 2(\Vert a\Vert^2 + \Vert b\Vert^2)$?2012-06-09
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    Or you want $\|a - b\| \leq \|a\| + \|b\|$? Or, taking squares, $\|a - b\|^2 \leq \|a\|^2 + \|b\|^2 + 2\|a\|\cdot\|b\|$.2012-06-09
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    Thanks for your replies. I realize that my initial question was incorrect, so I modified it.2012-06-14

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