Expected $f$vector of the Poisson Zero Polytope and Random Convex Hulls in the HalfSphere
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 $\mathbb R^d$. The expected $f$vector is expressed through the coefficients of the polynomial $$ (1+ (d1)^2x^2) (1+(d3)^2 x^2) (1+(d5)^2 x^2) \ldots. $$ 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 halfsphere. In the case when $n=d+2$, we compute the probability that this spherical convex hull is a spherical simplex, thus solving an analogue of the Sylvester fourpoint problem on the halfsphere.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 arXiv:
 arXiv:1901.10528
 Bibcode:
 2019arXiv190110528K
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 Mathematics  Metric Geometry;
 Primary: 52A22;
 60D05;
 Secondary: 52B11;
 52A20;
 51M20;
 52A55
 EPrint:
 29 pages, 2 figures, 5 tables. Major changes compared to the previous version. A formula for the "ugly" values of the expected $f$vector added. This is a preprint version containing tables