# Convex hull of $k$ random points

Suppose we have $k$ realizations of a random variable uniformly distributed over the unit cube $[0,1]^n$.

What is the probability that their convex hull has all of the $k$ points as extreme points?

If it would be easier, "unit cube" can be replaced by "unit ball".

• It seems to me that uniform probability on the unit cube is a bit artificial. The question is perhaps more natural for points chosen with respect to a probability which is invariant under all linear isometries, say uniform probability in a ball or (perhaps even better) points chosen with respect to a product of Gaussian measures. Sep 3, 2010 at 11:05
• @roland: i agree that spherically symmetric probability measures, such as the two you mention, seem more esthetic. if such a version of the problem interests you - go for it! (the problem i quoted is what was originally posted on yahoo.) Sep 3, 2010 at 16:57

Imre Bárány has investigated similar questions, including the asymptotics of $$p(k,S)$$, the probability that $$k$$ uniformly chosen points from the convex body $$S\subset \mathbb{R}^n$$ are in convex position (they are extreme points of their convex hull). In general one can give the bounds $$c_1\le k^{2/(n-1)}\sqrt[k]{p(k,S)}\le c_2$$ for large enough $$k$$ and constants $$c_1,c_2$$. I don't think closed form formulas are known for all $$k$$ even for simple convex sets $$S$$. See here and the papers in the references. See here for the case when $$S$$ is the unit ball.
The number of $$k$$-dimensional faces $$f_k$$ on a random polytope is well studied subject, and you are asking about the $$k=0$$ case. The distributions that have probably received the most attention are uniform distributions on convex bodies and the standard multivariate normal (Gaussian) distribution. As Gjergji mentioned, Bárány has some of the strongest results in this area. In particular Bárány and Vu proved central limit theorems for $$f_k$$.