Suppose we randomly pick $N$ points inside the unit hypercube in $\mathbb{R}^n$, and form their convex hull. What is the expected value of the volume of the convex hull?
For example, in the case $N=n=2$, we are asking about the average length of a line segment in a unit square. This can be shown to be $\frac{2+\sqrt{2} + 5\ln (1+\sqrt{2})}{15} \approx 0.521$.
Is there a method to compute this expected value in general?
Edit: as was made clear to me by a comment, I should clarify that the volume I am asking for is not coming from Lebesgue measure on $\mathbb{R}^n$, but rather $\mathbb{R}^{N-1}$.