I was reading a question on this website where OP asked some question on MC evaluation of the area of a circle. I have been thinking about a rather unrelated problem, which although of little practical use I thought could be interesting. So I am sharing it here...
Consider a closed centred ball $B$ in $\mathbb R^n$, whose radius $R>0$ is unknown. Let $(X_i)$ be a sequence of points drawn independently and uniformly from $B$. Fix $r>0$, and denote as $b_r(x)$ the closed ball centred in $x$ with radius $r$. Now, let $S_n(r) = \bigcup_{i=1}^nb_r(X_i)$, and define the random variable $$N(r) = \inf\{n\geq 1\;:\;S_n(r)\supseteq B\}\,.$$ It should not be too hard to show that $N(r)$ is finite almost surely. Only having access to one (or multiple independent) realisations of $N(r)$ for a fixed known $r$, what can we say about the volume of $B$? Note that we do not have access to the sequence of points, only to the value of $r$ and of $N(r)$.
Clearly, the volume of $B$ is going to be less than $N(r)V(r)$ where $V(r)$ is the volume of a ball or radius $r$ in $\mathbb R^n$. But more can be said... Note that already the case $N(r)=1$ is non-trivial. For sure, $R\leq r$. However, we are interested in high probability upper bounds for $R$. If $2R<r$, we have that $N(r)=1$ with probability $1$. The case $r\in[R, 2R]$ is more interesting. Let $r = R+\varepsilon$, with $\varepsilon\in(0,R)$. Then, $b_r(X_1)\supseteq B$ if, and only if $X_1\in b_\varepsilon(0)$. This happens with probability $(\varepsilon/R)^n$.
Given $R$ and $r$, the minimal value of $N(r)$ is the covering number of a ball of radius $R$ with balls of radius $r$. However since the sequence $(X_i)$ is random, $N(r)$ Will usually be much larger.
A related question would be what can be said if one has access to $N(r)$ for every value $r>0$.
