Given $n$ points placed uniformly and at random in a disk/square, on average how many convex hulls can be drawn if, every time the convex hull is drawn, every point on that convex hull is deleted?
Note that the convex hull of one or two points is just the set containing that/those one/two points. I was idly drawing and connecting up dots in one of my notebooks when this problem (or pair of problems, technically) came to me. I've done a little (seemingly largely ineffective) work, but don't know much about the math of convex hulls and so am not even certain I applied what I have correctly.
A quick bit of googling reveals that "the expected number of vertices of the convex hull of n points, chosen uniformly and independently from a disk is $O\big(n^{\frac{1}{3}}\big)$" and that "$O(k\log(n))$ for the case of a convex polygon with $k$ sides", which in my case with $n=4$ reduces to $O(4\log(n))$. If (taking for example the case of the disk), deleting the points in the first convex hull, we are left with $n-O\big(n^{\frac{1}{3}}\big)$, then it seems to me (at least as far as I understand it based on my limited understanding of big-$O$ notation) we can approximate the total number of convex hulls that can be drawn, particularly as $n\to\infty$, as $$\frac{O\big(n^{\frac{1}{3}}\big)}{1-\frac{n-O\big(n^{\frac{1}{3}}\big)}{n}}=\frac{n\cdot O\big(n^{\frac{1}{3}}\big)}{n-\big(n-O\big(n^{\frac{1}{3}}\big)\big)}=\frac{n\cdot O\big(n^{\frac{1}{3}}\big)}{O\big(n^{\frac{1}{3}}\big)}=n\cdot O(1)=O(n)$$ Which seems unintuitive to me. But this analysis, presumably amongst other errors, neglects that there is no reason as far as I can discern to believe that the points remaining after the removal of those in the previous convex hull on average behave as though uniformly distributed in some smaller disk for the purpose of using the same $O\big(n^{\frac{1}{3}}\big)$ figure for number of points in the convex hull of those remaining points mentioned in the paper above.
I haven't taken any classes that have covered big-$O$ notation or convex hulls in really any detail, so my limited knowledge of these fields almost exclusively stems from Wikipedia and various math videos on YouTube. So assuming I've made mistakes, that's likely to be why. I'm ideally hoping for someone to give a formal solution, explain where my analysis went wrong and suggest potentially fruitful avenues for solution, or something else along those general lines. But really anything would be appreciated—thanks!