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

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull.

• For what $N$ do we know the exact values of $P(N)$, $A(N)$?
• Is there a general formula for $P(N)$ or $A(N)$?
• What is the asymptotic behavior of $P(N)$ and $A(N)$ as $N\to\infty$?

$N$ points are selected in a uniformly distributed random way in a ball of the unit radius. Let $S(N)$, $V(N)$ denote the expected surface area and the expected volume of their convex hull.

• For what $N$ do we know the exact values of $S(N)$, $V(N)$?
• Is there a general formula for $S(N)$ or $V(N)$?
• What is the asymptotic behavior of $S(N)$ and $V(N)$ as $N\to\infty$?
• A good search keyword for this topic is "random polytopes". For example: Buchta and Müller, "Random polytopes in a ball", J. Appl. Prob. 21 (1984), 753–762, jstor. – user21467 May 10 '14 at 1:06
• Did you try simulating this problem? Multiple run might give you some patterns behind. – MFSO1991 Sep 17 '14 at 23:20

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.