EXPECTEDf‐VECTOR OF THE POISSON ZERO POLYTOPE AND RANDOM CONVEX HULLS IN THE HALF‐SPHERE

Abstract

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 R d . The expected f-vector is expressed through the coefficients of the polynomial ( 1 + ( d − 1 ) 2 x 2 ) ( 1 + ( d − 3 ) 2 x 2 ) ( 1 + ( d − 5 ) 2 x 2 ) … . 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.