Consider the bounded region given by $A \in \mathbb{R}^n$. Let it be given that we have uniformly sampled $k$ i.i.d. points ${P_i} \sim U(A)$ where $k$ is some constant.
Now $S \subseteq A$ be any convex region in $A$ s.t. forall $i$, $P_i \not\in S$. Then can we give an upper bound on the expected value of $vol(S)$ in terms of $vol(A)$, $k$, and $n$ ?
If this is not possible, what additional conditions may we assume/change on the region A and the points, so that we can come up with an upper bound for $vol(S)$?