The convex hull of random points in a tetrahedron: Solution of Blaschke's problem and more general results

摘要

Write vol(C) for the volume of th three-dimensional convex body C and [x_1, …, x_n] for the volume of the convex hull of the points x_1, …, x_n ∈ C. Then V~((i))(n, C) = 1/(vol(C)~i) ∫_C … ∫_C[x_1, …, x_n]~i (dx_1)/(vol(C)) … (dx_n)/(vol(C)) is the ratio of the i-th moment of the volume of the convex hull of n random points in C to the i-th power of the volume of C. In particular, V~((1))(4, C) is the ratio of the expected volume of a random tetrahedron in C to the volume of C.