# What is the expected size of the convex hull of $n$-points selected randomly in a $2d$-circle?

We know $n>2$ and worst case its a triangle, best case the points all lie on a circle. Can we generalize to higher dimensions? What's the probability that the size of the convex hull of $n$ points randomly selected in a circle is $k$?

The expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from a disk, is $O(n^{\frac{1}{3}})$.