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In the middle of the last century, J. J. Sylvester tried to calculate the expectation value of the convex hull of n randomly chosen points in a plane square. As he discovered, for n = 1, the problem is trivial. For n = 2, the question is relatively easy to answer with Mathematica. But starting with n = 3 the problem becomes extremely difficult due to the large number of integrals to be evaluated. ... The general setting of the problem is to evaluate the expectation value of the min(n-1, d)-dimensional volume of the convex hull of n points in d dimensions; for instance, the volume of a random tetrahedron formed by four randomly chosen points in R3 (an unsolved problem). In the following, we will show that, using the stunning integration capabilities of Mathematica, it is now possible to tackle such problems directly.
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