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 $m\rightarrow\infty$?
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$\begingroup$ I'm puzzled by how you could think this would go to $0$. You are asking whether the average distance between points must go to $0$ when you have more and more of them... of course not. $\endgroup$– Douglas ZareSep 23, 2013 at 22:33
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$\begingroup$ I can't even deduce a precise meaning and you already closed the question? Wasn't it better first to be a bit patient to find out what the question really was? $\endgroup$– Włodzimierz HolsztyńskiOct 6, 2013 at 3:02
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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 $t_m/m^2\simeq m(2m-1)\bar{t}/m^2$ will remain nonzero as $m\rightarrow\infty$.
answered Sep 23, 2013 at 5:17