Cones generated by random points on half-spheres and convex hulls of Poisson point processes
Cones generated by random points on half-spheres and convex hulls of Poisson point processes
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DOI：
10.1007/s00440-019-00907-3
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发表时间：
2018-01
影响因子：
2
通讯作者：
Z. Kabluchko;A. Marynych;Daniel Temesvari;C. Thäle
中科院分区：
文献类型：
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作者：
Z. Kabluchko;A. Marynych;Daniel Temesvari;C. Thäle
设为从d维上半球均匀独立地采样的随机点。我们证明了，在没有任何正规化的情况下，由弱收敛到某个极限随机向量的维凸锥的f-向量。我们还表明收敛的f-向量的所有时刻，并确定的期望的限制常数。我们证明了期望的格拉斯曼角可以用期望的f-向量表示。这产生了预期的格拉斯曼角和圆锥固有体积的收敛，从而回答了Bárány等人的问题。（随机结构算法50（1）：3-22，2017。 https://doi.org/10.1002/rsa.20644 ).我们的方法是基于这样的观察，即随机锥弱收敛，经过适当的重新缩放，到一个随机锥，其与切超平面的半球在其北极是凸船体的泊松点过程与幂律强度函数成比例，其中。我们计算了任意随机凸船体的期望面数、期望内禀体积和期望T-泛函。
Letbe random points sampled uniformly and independently from thed-dimensional upper half-sphere. We show that, as, thef-vector of the-dimensional convex conegenerated byweakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of thef-vector ofand identify the limiting constants for the expectations. We prove that the expected Grassmann angles ofcan be expressed through the expectedf-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644 ). Our approach is based on the observation that the random coneweakly 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 of the Poisson point process with power-law intensity function proportional to, where. We compute the expected number of facets, the expected intrinsic volumes and the expectedT-functional of this random convex hull for arbitrary.