Suppose we randomly pick $N$ points inside the unit hypercube in $\mathbb{R}^{N}$. What is the expected value of the volume of the convex hull (in any $\mathbb{R}^{N-1}$ containing the convex hull)?

For example, if $N=2$, 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?

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    $\begingroup$ In addition to Gery Myerson's comments: For $N=2$ you would have two points in $[0,1]$ and not in the unit square. Probably you want $N$ random points in $[0,1]^N$? $\endgroup$ – Dirk Sep 25 '16 at 0:46
  • $\begingroup$ Apologies - edits added for clarity. $\endgroup$ – user38495 Sep 25 '16 at 4:54
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    $\begingroup$ "The average area of a triangle picked at random inside a unit cube is $A=0.15107 \pm 0.00003.$" See this link. $\endgroup$ – Joseph O'Rourke Sep 26 '16 at 1:12

This problem is answered (in the limit) by

M. E. Dyer, Z. Furedi, and C. McDiarmid. Random volumes in the n-cube. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 1:33–38, 1990;

and the specific question might well be answered in something that cites this.


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