Expected values of some properties of the convex hull of a random set of points

Solution 1:

My intuition of a combinatorial geometer suggests the following and that we shall have the typical situation. Exact values of $P(N), A(N), S(N)$, and $V(N)$ can be calculated for small $N$ and the compexity of these calculations grows very quickly with $N$, so may be we shall be able to formulate a right conjecture for a general exact formula of $P(N), A(N), S(N)$, and $V(N)$, but we shall not be able to prove it. From the other side, my intuition suggests that when $N$ tends to the infinity then $P(N)$ tends to perimeter of the unit disk, $A(N)$ tends to area of the unit disk, $S(N)$ tends to surface area of the unit ball, and $V(N)$ tends to volume of the unit ball, but I cannot say now, how quickly.