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Let $\{z_i\}$, $i=1,2,\ldots,n$ be a set of complex numbers. Then I know that there is a set $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{\pi} \sum_{k=1}^n |z_k|. $$ However, how do I show that there's some other set, say M such that $$\left|\sum_{j\in M} z_j\right|\ge \frac{1}{8} \sum_{k=1}^n |z_k|. $$

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    Won't the same set work? Or do you need it to be distinct from $J$?2012-01-25
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    That's what I thought, but I don't know how to get the 1/8.2012-01-25
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    Show that $8 \ge \pi$. btw, what is the source of this problem?2012-01-25
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    Related post: http://math.stackexchange.com/q/91939/13425. (@Aryabhata: Pinging you since I think you might be interested in that post.)2012-01-25
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    @Srivatsan: Thanks for the link!2012-01-25
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    @Aryabhata: Thanks. Is there another way of doing it without using the first inequality?2012-01-25
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    @Joel: See the related post Srivatsan referred to. I ask again: Do you know the source? Knowing that might clarify the problem a bit. (and it would help folks who want to read further).2012-01-25
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    @Aryabhata Sorry for the shameless plug, but in case you are interested: I posted [a $d$-dimensional generalisation of this question](http://math.stackexchange.com/q/102508/) today. Let's hope that it turns up something more in the way of motivation.2012-01-26
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    @Srivatsan: I am interested! You should not feel sorry for sharing :-)2012-01-26

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