Consider the unit $n$-sphere and $l$ points distributed uniformly randomly inside its hypervolume. What is the expectation of the smallest distance from a point to the sphere centre?
Steps done so far are: Find distribution function of distance from random point to zero: $P(\xi < x) = x^n$, and the distribution function of the closest to zero point is $P(\xi < x) = 1 - (1-x^n)^l$. Then considering E(x) = $\int_{-\infty}^{+\infty} x \cdot d(F(x))$, the problem is how to calculate it.
The cdf of the random variable describing the distance $r$ from a random point in the unit $n$-sphere to the centre is $r^n$, so the pdf is its derivative or $nr^{n-1}$. Now we are looking for the first order statistic of a sample of $l$ points from this distribution, whose pdf computes to (using the formula given on MathWorld) $$l(1-r^n)^{l-1}nr^{n-1}$$ Thus the desired expectation is $$E(n,l)=\int_0^1rl(1-r^n)^{l-1}nr^{n-1}\,dr$$ Substitute $s=r^n$: $$=l\int_0^1s^{1/n}(1-s)^{l-1}\,ds=l\mathrm B\left(1+\frac1n,l\right)=\frac{l\Gamma(1+1/n)\Gamma(l)}{\Gamma(1+1/n+l)}=\frac{l!}{\prod_{i=0}^{l-1}(1+1/n+i)}$$ $$E(n,l)=\prod_{i=1}^l\frac i{1/n+i}$$