1. Given $n$ independent uniformly distributed points on $S^2$, what's the distribution of the distance between two closest points?
2. Consider $n$ iid uniform points on $S^1$, $Y\_1, \ldots, Y\_n$, in counterclockwise order. Now let $I\_1 = Y\_2-Y\_1, \ldots, I\_n = Y\_1 - Y\_n$ be the spacings between consecutive points. Finally order the spacing sequence into $I\_{(1)} < I\_{(2)} < \ldots < I\_{(n)}$. They will also generate a spacing sequence, of size $n-1$, $J\_1 = I\_{(2)} - I\_{(1)}, \ldots, J\_{n-1} = I\_{(n)} - I\_{(n-1)}$. What's the distribution of this last sequence? In particular, what's the mean value of the smallest $J$ and largest $J$?



Two geometric probability questions (one answered, one more to go)
There is an asymptotic formula for the minimal spherical distance when $n$ is large (see e.g. the PhD thesis [*"Random Diameters and Other U-Max-Statistics"*](http://www.imsv.unibe.ch/content/research/publications/theses/2008/e6079/e6261/e6950/Mayer2008_eng.pdf) by M. Mayer, Corollary 3.37):



> 
> **Theorem.** Assume that the points $\xi\_1,\xi\_2,...,\xi\_n$ are independent and uniformly distributed on $\mathbb S^{d−1}$. Let $S\_n$ be the smallest central angle formed by point pairs within the sample. Then for $t > 0$
>  $$P\{n^{2/(d-1)}S\_n\leq t\}=1-\exp\left(-\frac{\Gamma(\frac{d}{2})}{4\pi^{1/2}\Gamma(\frac{d+1}{2})}t^{d-1} \right)+\mathcal O(n^{-1}).$$
> 
> 
> 


I am not sure if there is a nice explicit formula for finite $n$. 


In fact, the knowledge of the exact form of the distribution $P\{S\_n\leq\theta\}$ on $\mathbb S^2$ would lead to a solution of the [Tammes packing problem](http://en.wikipedia.org/wiki/Tammes_problem) (which is only solved for a few values of $n$ to the best of my knowledge).


