Say I construct a set of points $S$ by drawing $k$ points from $\mathbb{R}^{n}$ independently from say a gaussian distribution*. I am wondering how the expected size of the set $E$ consisting of the extreme points of $S$ grows as a function of $k$?
I am not sure I am using the right terminology here, but by extreme points I mean the points in $S$ that are on the edge of the convex hull of $S$. I.e., if we imagine a rubber band around the convex hull of $S$, $E$ are the points of $S$ touching this rubber band. I hope that make sense.
Anyway, my intuition is that the expected size of $E$ only grows very slowly in $k$, say something like $O(\log(k))$. Does anyone know if that is actually the case?
*I am trying to reason about some naturally occuring data, essentially the points are lumped together around some mean point, so a gaussian seems a natural approximation.