TITLE: What is the expected convex depth of a set of $m$ randomly chosen points in the unit square?
QUESTION [11 upvotes]: Definition.  Let $X$ be a set of points in $\mathbb{R}^2$.  Then the vertex sequence of $X$ is defined by

$X_{0}=X$, and
$X_{i+1}=\left\{x\in \operatorname{Conv}(X_{i}): x \notin \operatorname{Conv}(X_{i}\setminus\{x\})\right\}$.

The convex depth of $X$ is the smallest $k\in \mathbb{N}$ such that $X_{k}$ is empty.

Here is an example where $|X|=30$.  
My question is,

What is the expected convex depth of a set of $m$ randomly chosen points in the unit square?

The answer should be a function of $m$.  I have simulated some data for this, pictured below, where the $x$- axis is $m$ and $y$- axis is the mean convex depth.  It looks $\sqrt{m}$ish (or maybe something closer to $m^{3/2}$, as suggested by lhf's link in the comments).  I'm not sure how to prove it, though.
.

REPLY [3 votes]: It's $m^{2/3}$. See Ketan Dalal, Counting the onion, Random Struct. Alg., 24 (2004), 155–165, doi:10.1002/rsa.10114.