Let $t_{m}$ be the sup of the sum of the pairwise distances between any $2m$ points in the unit disk. Does $t_{m}/m^{2}$ go to $0$ as $m\rightarrow\infty$?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
closed as offtopic by Benoît Kloeckner, Ramiro de la Vega, Ricardo Andrade, Suvrit, Kevin P. Costello Sep 23 '13 at 22:09This question appears to be offtopic. The users who voted to close gave this specific reason:



No, it need not go to $0$: just put $m$ points at $(0,1/2)$ and the other $m$ points at $(0,+1/2)$; the sum of the pairwise distances is $t_m=m^2$, so $t_m/m^2=1$ does not vanish as $m\rightarrow\infty$. More generally, if you distribute the points inside the unit disk with some generic density function, there will be some nonzero average pairwise distance $\bar{t}$, and $t_m/m^2\simeq m(2m1)\bar{t}/m^2$ will remain nonzero as $m\rightarrow\infty$. 

