Random Gaussian Tetrahedra

Abstract

Given independent normally distributed points A,B,C,D in Euclidean 3-space, let Q denote the plane determined by A,B,C and D^ denote the orthogonal projection of D onto Q. The probability that the tetrahedron ABCD is acute remains intractable. We make some small progress in resolving this issue. Let Gamma denote the convex cone in Q containing all linear combinations A+r*(B-A)+s*(C-A) for nonnegative r, s. We compute the probability that D^ falls in (B+C)-Gamma to be 0.681..., but the probability that D^ falls in Gamma to be 0.683.... The intersection of these two cones is a parallelogram in Q twice the area of the triangle ABC. Among other issues, we mention the distribution of random solid angles and sums of these.