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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1Won't the same set work? Or do you need it to be distinct from $J$? – Aryabhata Jan 25 '12 at 18:59
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That's what I thought, but I don't know how to get the 1/8. – Joel Jan 25 '12 at 19:00
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4Show that $8 \ge \pi$. btw, what is the source of this problem? – Aryabhata Jan 25 '12 at 19:01
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1Related post: http://math.stackexchange.com/q/91939/13425. (@Aryabhata: Pinging you since I think you might be interested in that post.) – Srivatsan Jan 25 '12 at 19:04
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@Srivatsan: Thanks for the link! – Aryabhata Jan 25 '12 at 19:08
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@Aryabhata: Thanks. Is there another way of doing it without using the first inequality? – Joel Jan 25 '12 at 19:45
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1@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). – Aryabhata Jan 25 '12 at 20:01
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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 today. Let's hope that it turns up something more in the way of motivation. – Srivatsan Jan 26 '12 at 02:49
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@Srivatsan: I am interested! You should not feel sorry for sharing :-) – Aryabhata Jan 26 '12 at 05:41