Average vs extreme spacing on $N$ random points on the unit interval Given a small number $n$ (in my case, $n=2$) and a large number $N$, generate a set $S_N$ with $N$ points uniformly at random on $[0, 1)^n$. I'm looking for an asymptotic for the average largest gap between neighbors
$$
\max_{s\in S}\min_{t\in S}\|s-t\|
$$
as $N\to\infty.$ I assume the ratio of this to the average spacing diverges to $\infty$ for all $n\ge1$, is that right?
Actually, even better (a more direct answer to my underlying question) would be: what is the largest $r$ such that there is a point $x\in[0,1)^n$ such that the ball of radius $r$ centered at $x$ does not contain any point in $S$:
$$
B_r(x)\cap S=\emptyset.
$$
Of course I'm looking for some kind of an average over all random choices of $S$ and taking $N\to\infty$ as before.

Assuming we are in the unit square, for each such point $x$, where $r$ is less than the distance from the boundary of the unit square*, the probability that there is no point in the interior of $B_r(x)$ is given by $(1-\pi r^2)^N$.  As $N$ goes to infinity, the probability goes to zero for all positive $r$.  So the largest such $r$ is $0$.  This answer holds regardless of dimension.
*For points closer to the boundary, you can throw in a factor of $1/4$ to bound the result and get the same answer.
