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Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $$ \min_{1\le i\ne j\le d}\left(x_{i}+x_{j}-\sqrt{x_{i}^{2}+x_{j}^{2}+2x_{i}x_{j}\cos\left(y_{i}-y_{j}\right)}\right)\le\frac{80}{d^{3}} $$ hold?

Thanks for any helpful answers.

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    I don't know either.2012-11-10
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    What have you done? Where are you stuck?2012-11-10
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    @Pragabhava, I have checked it for more than a hundred sets of real numbers that satisfy the assumption. I have also looked at the $y$'s that only lie in $[0,\pi]$ but couldn't get it.2012-11-11
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    Define $\omega_k = e^{i t_k}$. Then the square root in your expression is precisely $|\omega_i^2 x_i^2 + \omega_j^2 x_j^2|$. Could be useful.2012-11-11
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    [This question](http://math.stackexchange.com/questions/234728) has been closed as a duplicate of the present one even though it had the stronger bound $80$ instead of $16\pi^2$. Please explain why you posed the question twice in two different guises and with different bounds. While you're at it, you might want to explain how you came up with these bounds; that might provide more motivation for proving them.2012-11-11
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    All the areas of math seem to be connected. A problem that may not be solved in one area could be solved in another area. Different approaches in different areas and different experiments generated different bounds. In one experiment, if one uses different parameters, one can also get different results.2012-11-13

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