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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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    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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