| A 3.339
Vortrag: Angles of random simplices
Abstract: The angle sum of any triangle in the plane is constant and equals half of the full plane angle. In dimension 3, the sum of the three-dimensional solid angles of a tetrahedron is not constant. It therefore makes sense to ask what is the "mean" angle sum of the tetrahedron. The answer depends on the probability measure we put on the set of tetrahedra. In this talk we shall show that if the four vertices of the tetrahedron are sampled uniformly from the unit sphere, then the expected sum of three-dimensional solid angles at its vertices equals 1/8 of the full solid angle. For vertices sampled uniformly from the unit ball, the result is 401/2560 of the full solid angle. These examples are special cases of a general theory dealing with simplices whose vertices are sampled from the multidimensional beta distributions. We shall exlain how the angles of these simplices are related to various models of stochastic geometry such as Poisson hyperplane tessellations, Poisson-Voronoi cells, random polytopes in convex bodies and on the half-sphere.