${{\mathbb{R}}^{n}}$ 中凸集的汇聚概率

${{\mathbb{R}}^{n}}$ 中凸集的汇聚概率

The Aggregation Probability of Convex Sets in ${{\mathbb{R}}^{n}}$

摘要/Abstract

参考文献 17

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数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 619-629.

严彤(),邹都^*()

武汉科技大学理学院武汉 430081

收稿日期:2023-05-22 修回日期:2024-10-15 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 邹都 E-mail:2771704373@qq.com;zoudumath@wust.edu.cn
作者简介:严彤，E-mail:2771704373@qq.com
基金资助:
国家自然科学基金(12171378);湖北省自然科学基金(2020CFA079)

Tong Yan(),Du Zou^*()

college of science, Wuhan University of Science and Technology, Wuhan 430081

Received:2023-05-22 Revised:2024-10-15 Online:2025-04-26 Published:2025-04-09
Contact: Du Zou E-mail:2771704373@qq.com;zoudumath@wust.edu.cn
Supported by:
NSFC(12171378);Hubei Natural Science Foundation(2020CFA079)

摘要：

利用 ${{\mathbb{R}}^{n}}$ 中凸体与平坦凸体两者平均曲率积分之间的关系, 给出 $h$ 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. 在此基础上得到了在凸体 $K$ 内有一个半径为 $r$ 而和所有随机平面有公共点的球存在的概率, 并进一步讨论了一维和二维情形下点的汇聚概率.

关键词: 平均曲率积分, 平行凸集, 几何概率, 均质积分, Minkowski 和

Abstract:

Based on the relationship between the average curvature integrals of the convex and flat convex bodies in ${{\mathbb{R}}^{n}}$ , the geometric probability that $h$ linear subspaces intersecting the convex body $K$ within $K$ is given. On this basis, the probability of the existence of a sphere with a radius of $r$ and a common point with all random planes in the convex body $K$ is given. Furthermore, the convergence probabilities of the points in the one-dimensional and two-dimensional cases are discussed.

Key words: mean curvature integral, parallel convex sets, geometric probability, homogeneous integration, minkowski addition

中图分类号:

O168.5

严彤,邹都. ${{\mathbb{R}}^{n}}$ 中凸集的汇聚概率[J]. 数学物理学报, 2025, 45(2): 619-629.

Tong Yan,Du Zou. The Aggregation Probability of Convex Sets in ${{\mathbb{R}}^{n}}$ [J]. Acta mathematica scientia,Series A, 2025, 45(2): 619-629.

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