I am trying to develop an intuition for the properties of the convex hull of a set of points in high ($d>20$) dimensions.
Consider a set of $n$ data points which are iid distributed according to some simple distribution (e.g. uniform hypercube, multivariate normal with mean $\mathbf{0}$ and identity covariance matrix $\mathbf{I}_d$, or similar). Now suppose we draw an $n+1$'th data point $\mathbf{x}$ from that same distribution. What can we say about the relationship between $\mathbf{x}$ and the convex hull of the first $n$ points?
My expectation$^1$ is that as dimensionality increases, the distance between $\mathbf{x}$ and the convex hull should grow for fixed $n$. I also expect the volume of the convex hull to, in some sense, shrink relative to the volume of the domain or something similar.
To make this more concrete, I would like some expression (or bound) on the expected distance between $\mathbf{x}$ and the convex hull for some simple data distribution as a function of $n$ and $d$.
Does such an example exist (either analytic or numeric) which could aid with my intuition?
Disclaimer: this is not my area of expertise, so even simple examples or half-answers would be very helpful.
$^1$ Inspired by section 2.5 of The Elements of Statistical Learning where the authors demonstrate the curse of dimensionality (e.g. points tend to be further apart as dimensions increase, side-length of the subcube needed to capture a fraction r of the volume of the data increases with dimension).