Probability that randomly chosen balls have a nonempty common intersection

Question

Fix some $0 < r < 1$. A collection of points $x_1, \dots, x_n$ are chosen independently and uniformly at random from the closed unit ball in $\mathbb R^d$.

What is the probability that the intersection $\cap_{i = 1}^n B_r (x_i)$ is nonempty?

Note: Here $B_r (x_i)$ is the closed ball of radius $r$ around $x_i$.

Comments: The question as posed is rather open ended since there may not be a simple expression for the probability. If an exact solution is hard or impossible to find, I would still be interested in asymptotic/approximate results.

Even the case $d = 1$ seems to be interesting, and may admit a reasonably simple exact expression.

Answer 1

The case $d=1$ is simple. Then the probability in question is $$p:=P(V-U\le r),$$ where $U$ and $V$ are, respectively, the smallest and largest order statistics for an iid random sample of size $n$ from the uniform distribution over $[0,1]$. The joint pdf of $(U,V)$ is given by $$f(u,v)=n(n-1)(v-u)^{n-2}\,1(0<u<v<1)$$ (assuming $n\ge2$). So, $$p=\int_0^1\int_0^1 du\,dv\,f(u,v)\,1(v-u\le r) =\begin{cases} 1&\text{ if }r>1,\\ r^{n-1}(n+r-nr)&\text{ if }r\in[0,1]. \end{cases} $$