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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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    Thanks for your replies. I realize that my initial question was incorrect, so I modified it.2012-06-14

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No. Let $a \neq 0$ and $b = -a$. The left hand side is $4\|a\|^2$ and the right hand side is $2\|a\|^2$.