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Let $A_1, A_2,\ldots, A_n$ be $n$ points on the unit circle centered at $O$. It is known that $\overline{OA_1}+\overline{OA_2}+\ldots +\overline{OA_n}=\overline{0}$.

Define the following two quantities $$S_1=\frac{1}{n}\sum_{j=2}^n A_1A_j$$ and $$S^{*}=\frac{1}{n^2}\sum_{1\le i<j\le n} A_iA_j.$$

Here $A_iA_j$ denotes the Euclidean distance between $A_i$ and $A_j$.

Can one prove that $S_1+S^{*}\le 2$. (An even better bound may be possible).

[Edit: Fejes Tóth proved that if $n$ points lie on a unit circle then $S^∗\le\frac2\pi$, see

Fejes Tóth, L. On the sum of distances determined by a pointset. Acta Math. Acad. Sci. Hungar., 7, 1956, 397–401.

On the other hand the condition that the center of gravity is located at $O$ gives easily a bound $S_1\le\sqrt2$. (Just use the fact that $\sum_{j=2}^nA_1A^2_j=2n$ and Cauchy-Schwarz). From these two, one can get an upper bound $S_1+S^∗\le\sqrt2+\frac2\pi=2.0508\dots$ But one can notice that this is not tight since if $S_1$ is close to $\sqrt2$ then $S^∗$ is close to $1/2$. I did some numerical experiments which suggest $2.0508$ can be replaced by $2$.]

Thanks,

Dan

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    $\begingroup$ I have a few typos above. Fejes Toth proved that if $n$ points lie on a unit circle then $S^{*}\le \frac{2}{\pi}$. On the other hand the condition that the center of gravity is located at $O$ gives easily a bound $S_1\le \sqrt{2}$. (just use the fact that $\sum_{j=2}^n A_1A_j^2=2n$ and Cauchy_Schwarz). From these two one can get an upper bound $S_1+S^{*}\le \sqrt{2}+\frac{2}{\pi}=2.0508\ldots$. But one can notice that this is not tight since if $S_1$ is close to $\sqrt{2}$ then $S^{*}$ is close to $1/2$. I did some numerical experiments which suggest $2.0508$ can be replaced by $2$. $\endgroup$ – user84909 Jul 15 '13 at 0:55
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    $\begingroup$ Re the comment beginning "I have a few typos...." .... Why wasn't this in the original post? $\endgroup$ – Steven Landsburg Jul 15 '13 at 2:29
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    $\begingroup$ user84909: I was asking why the content of your "I have a few typos..." comment --- i.e. the information about what's known from Fejes Toth, the information about your numerical experiments, your observations using Cauchy Schwarz --- was not in the original post. I voted to close because you seem to have intentionally withheld this information. I believe this could be turned into a good question if you were more forthcoming. $\endgroup$ – Steven Landsburg Jul 15 '13 at 3:12
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    $\begingroup$ Also, "I challenge you to ..." No, no challenges here either. Please. $\endgroup$ – Andrés E. Caicedo Jul 15 '13 at 3:44
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    $\begingroup$ @user84909, instead of clicking the add comment button, consider clicking the edit button on the question. $\endgroup$ – Joel Reyes Noche Jul 15 '13 at 4:23

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