What is the probability that the convex hull of $n$ randomly distributed points has $l$ of the points on its boundary?
Consider a square in which $n$ points are uniformly randomly distributed. Now consider the convex hull of these points. The "length" of the convex hull is defined as the number of points in the perimeter of the convex hull i.e. $n \\; -$ the number of points strictly contained by the convex hull.
I conjecture that the probability that the convex hull has "length" $l$ is the same for all $3 \le l \le n$ as the points are randomly arranged in the grid. However, I’m unable to come up with a proof for this claim.
Is my conjecture correct, and if not, what is the correct probability distribution?
Note that the points do not lie on any grid - there are infinite possibilities for the arrangement of points and (almost surely) there will be no identical points.