Cones generated by random points on half-spheres 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 half-sphere. We show that, as \(n\rightarrow \infty \), 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 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 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 of the Poisson point process with power-law intensity function proportional to \(\Vert x\Vert ^{-(d+\gamma )}\), 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 \(\gamma >0\).

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Acknowledgements

We would like to that the referee, whose comments helped us to improved our text. The work of AM was supported by the return fellowship of the Alexander von Humboldt foundation. DT was supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-Dimensional Phenomena in Probability—Fluctuations and Discontinuity. ZK and CT were supported by the DFG Scientific Network Cumulants, Concentration and Superconcentration.

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Correspondence to Alexander Marynych.

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Kabluchko, Z., Marynych, A., Temesvari, D. et al. Cones generated by random points on half-spheres and convex hulls of Poisson point processes. Probab. Theory Relat. Fields 175, 1021–1061 (2019). https://doi.org/10.1007/s00440-019-00907-3

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Keywords

  • Blaschke–Petkantschin formula
  • Conic intrinsic volume
  • Convex cone
  • Convex hull
  • f-Vector
  • Random polytope
  • Poisson point process
  • Spherical integral geometry

Mathematics Subject Classification

  • Primary 52A22
  • 60D05
  • Secondary 52A55
  • 52B11
  • 60F05