I am wondering if any convex geometers/probabilists have looked at the following question:

Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each point $x_i\in\{x_1,\ldots, x_n\}$, draw $N$ points uniformly on the sphere $S^{d-1}$ with radius $r>1$ centered at $x_i$, denote as $x_{i, 1}, \ldots, x_{i, N}$. Let $C$ be the convex hull of $x_{1,1}, \ldots, x_{1,N},\ldots, x_{n,1},\ldots,x_{n,N}$. What is the probability that $\forall i\in\{1,\ldots, n\}$, $B(x_i, 1)\in C$?

So in other words, for every original point $x_i$, we draw a unit ball around it, how likely that $C$ contains all these unit balls?

I found Probability that a convex shape contains the unit ball was asking a similar question. According to the comments, if $N$ is exponential in $d$, then my question holds with probability $1$, because for each $i$, the convex hull of $x_{i, 1},\ldots, x_{i, N}$ already contains a unit ball. My question is different that we have $n$ points, and I imagine the neighboring vertices help each other to enlarge the convex hull. So perhaps a tighter bound exists?

Has this problem been studied before? What are the assumptions that people put on the distribution of $x_i,\ldots, x_n$? Thanks for any comments/answers!

$\textbf{Update:}$ what if we make the original n points to be d+1 points that make a regular $d$-simplex?