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Cones generated by random points on half-spheres and convex hulls of Poisson point processes. (arXiv:1801.08008v2 [math.PR] UPDATED)
来源于:arXiv
Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently
from the $d$-dimensional upper half-sphere. 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 B\'ar\'any, 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 half-sphere at its north pole is the convex hull
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