Cones generated by random points on halfspheres and convex hulls of Poisson point processes
Abstract
Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$dimensional upper halfsphere. We show that, as $n\to\infty$, the $f$vector of the $(d+1)$dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the halfsphere at its north pole is the convex hull of the Poisson point process with powerlaw intensity function proportional to $\x\^{(d+\gamma)}$, where $\gamma=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$functional of this random convex hull for arbitrary $\gamma>0$.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.08008
 Bibcode:
 2018arXiv180108008K
 Keywords:

 Mathematics  Probability;
 Mathematics  Metric Geometry;
 52A22;
 60D05 (Primary) 52A55;
 52B11;
 60F05 (Secondary)
 EPrint:
 31 pages, 2 figures