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
