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Given x and y, is there any way we can express $||x|| + ||y|| - ||x+y||$ in terms of $||y-x||$? Even a bound where $||x|| + ||y|| - ||x+y|| \leq f(||y-x||)$ for some $f(\cdot)$ would be desirable.

Geometrically, ||x|| and ||y|| could be two sides of a parallelogram and then $||x+y||$ and $||y-x||$ would be its diagonals.

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  • $\begingroup$ I will put back the way the quesiton was originally, and then ask a new question. $\endgroup$ – MotiNK Nov 18 '15 at 10:06
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$$||x||+||y||-||x+y|| \\ = ||x||-\frac{||x+y||}{2}+||y||-\frac{||x+y||}{2} \\ \le ||x -\frac{x+y}{2}|| + ||y -\frac{x+y}{2}|| \\ \le \frac{||x-y||}{2} + \frac{||y-x||}{2} \\ \le ||y-x||$$

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  • $\begingroup$ Very nice! I just want to comment that the last two $\leq$ could be changed to $=$. It is not wrong what you wrote, but having only one $\leq$ will point out more clearly (in my opinion) what is happening. $\endgroup$ – mickep Nov 18 '15 at 11:12
  • $\begingroup$ I left the $\le$ not to confuse the reader if he only reads first and last line $\endgroup$ – stity Nov 18 '15 at 11:19

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