$\mathbb{R}^n$中线性子空间束与凸体相交的几何概率

数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 770-782.

$\mathbb{R}^n$中线性子空间束与凸体相交的几何概率

赵江甫()

^{福建江夏学院数理教研部福州 350108}

收稿日期:2019-09-18 出版日期:2021-06-26 发布日期:2021-06-09
作者简介:赵江甫, E-mail: 2833811196@qq.com
基金资助:
福建省中青年教师教育科研项目基金(科技类)(JT180585);福建江夏学院科研培育人才项目基金(JXZ2019016)

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

Jiangfu Zhao()

^{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)

摘要/Abstract

摘要：

利用超曲面的平均曲率积分的概念及性质给出$\mathbb{R}^n$中三个与凸体K相交的线性子空间彼此在K内相交的几何概率，并重点讨论超平面束的情形.在此基础上给出超平面束分别与特殊凸体：球体、正方体以及长方体相交时的几何概率，并讨论这些概率序列的单调性、收敛性以及大小关系.

关键词: 平均曲率积分, 几何概率, 蒲丰投针, 初等对称函数, 超平面束, 线性空间束

Abstract:

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

中图分类号:

O168.5

赵江甫. $\mathbb{R}^n$中线性子空间束与凸体相交的几何概率[J]. 数学物理学报, 2021, 41(3): 770-782.

Jiangfu Zhao. Geometric Probability of Subspaces Intersecting with a Convex Body in $\mathbb{R}^n$[J]. Acta mathematica scientia,Series A, 2021, 41(3): 770-782.

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