Tightly placed sphere caps

Question

Problem. Fix a positive real $r>0$ and a positive integer $n>0$. Consider an independent, identical sample $X_1, \ldots X_n$ drawn from the uniform distribution over the unit $d$-sphere $\mathbb S^d \subset \mathbb R^{d+1}$. What is the probability that $\|X_i - X_j\| \le r$ for every $(i,j)$?

I could work out an iterated integral giving an exact formula for $d=1$, although the integral itself is quite complicated. Before proceeding further with calculations, I wanted to know if there is a prior literature on this problem. This area of mathematics seems to go by the name of geometric probability (e.g. this 1978 textbook by Solomon).

A similar, well-studied problem is the probability that spherical caps of radii $r$ centered at $X_1, \ldots X_n$ covers the whole sphere. A spherical cap here refers to sets of the form $\{ y \in \mathbb{S}^d | \|x-y\| \le r \}$ for each fixed $x \in \mathbb S^d$. For this problem, an exact solution is known for $d=1$ (Stevens 1939), and some work on upper and lower bounds of the probability are known for $d>1$ (Gilbert 1965). The $d=1$ case by Stevens is as follows. The probability that $n$ randomly chosen arcs of length $a$ from a circle of circumference $1$ is equal to: $$\sum_{k = 0}^{\lfloor 1/a \rfloor} (-1)^k \binom nk (1- ka)^{n-1} $$

Answer 1

Not an answer; too long for a comment:

There seem to be extensive results in the paper https://arxiv.org/abs/0712.2816 Coverage processes on spheres and condition numbers for linear programming by Bürgisser et al. I am unsure whether they will help with your problem.

From the abstract:

Let $p(n,m,\alpha)$ be the probability that $n$ spherical caps of angular radius $\alpha$ in $S^m$ do not cover the whole sphere $S^m.$ We give an exact formula for $p(n,m,α)$ in the case $\alpha \in [π/2,π]$ and an upper bound for $p(n,m,\alpha)$ in the case $\alpha \in [0,π/2]$ which tends to $p(n,m,\pi/2)$ when $\alpha \rightarrow \pi/2.$

In the case $\alpha \in [0,\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius $\alpha$ that are needed to cover $S^m.$

The authors then relate the results to the condition number of a random linear program in $\mathbb{R}^{m+1}.$