EXPECTED f-VECTOR OF THE POISSON ZERO POLYTOPE AND RANDOM CONVEX HULLS IN THE HALF-SPHERE

摘要

We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in Rd. The expected f-vector is expressed through the coefficients of the polynomial (1+(d-1)2x2)(1+(d-3)2x2)(1+(d-5)2x2)....Also, we compute explicitly the expected f-vector and the expected volume of the spherical convex hull of n random points sampled uniformly and independently from the d-dimensional half-sphere. In the case when n=d+2, we compute the probability that this spherical convex hull is a spherical simplex, thus solving the half-sphere analogue of the Sylvester four-point problem.