Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (3): 770-782.

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Geometric Probability of Subspaces Intersecting with a Convex Body in $\mathbb{R}^n$

Jiangfu Zhao()   

  1. Department of Mathematics and Physics, Fujian Jiangxia University, Fuzhou 350108
  • Received:2019-09-18 Online:2021-06-26 Published:2021-06-09
  • Supported by:
    the Educational Research Project Fund of Young and Middle-Aged Teachers of Fujian Province(JT180585);the Project Fund for Scientific Research and Cultivation of Talents of Fujian Jiangxia University(JXZ2019016)


The probability that three independent random subspaces in $\mathbb{R}^n$ intersecting a convex body K have their common point intersecting K is found by using of the mean curvature integral of convex sets. Then we focus on the particular case of hyperplanes. On the base of this, we state the geometric probability of hyperplanes that intersect a ball, a cube or a right parallelepiped having an intersection inside the same ball, the cube or the right parallelepiped respectively. Finally, the monotonicity, convergence, and size relationship of the geometric probabilistic sequence are discussed.

Key words: Mean curvature integral, Geometric probability, Buffon needle throwing, Elementary symmetric function, Hyperplanes, Subspaces

CLC Number: 

  • O168.5