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Does the inequality $2 \langle x , y \rangle \leqslant \langle x , x \rangle + \langle y , y \rangle $, where $ \langle \cdot, \cdot \rangle $ denotes scalar product, have a name?

I've tried looking at several inequalities on wikipedia but I didn't find this one. And of course googling doesn't work for this purpose.

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    @BeniBogosel: Thank you for your wishes and for the answer :)2012-02-24

2 Answers 2

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It is the development of $ \langle x-y,x-y \rangle \geq 0$ and it follows from the positive definitness of the scalar product.


Apart of the above proof of the inequality, and as a response to the comments to the question, here are a few reasons as to why this inequality should be true, at a first glance:

  • a scalar product has the properties of the multiplication on the real line, so the inequality $2xy\leq x^2+y^2$ should pop up while looking at the given inequality;

  • Cauchy Schwarz immediatley implies the inequality: $2\langle x,y \rangle \leq 2 \|x\|\|y\| \leq\|x\|^2+\|y\|^2 $

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    I would not consider anyone who answered this question guilty. I would like to know the motivation of the OP for posting this question.2012-02-22
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It is essentially Young's inequality.

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    Okay, maybe that's stretching it. But it is not far from the truth.2012-02-22