R^(n)中的随机弦长矩不等式与随机弦长的分布函数  

Inequalities of the Moments of Random Chord Length and Distribution Functions of Random Chord Length in R^(n)

作　　者：赵江甫 蒋君[2,3] Jiangfu Zhao;Jun Jiang(Department of Mathematics and Physics,Fujian Jiangxia University,Fuzhou 350108,P.R.China;College of Science,Wuhan University of Science and Technology,Wuhan 430065,P.R.China;Hubei Province Key Laboratory of Systems Science in Metallurgical Process,Wuhan 430065,P.R.China)

机构地区：[1]福建江夏学院数理教研部,福州350108 [2]武汉科技大学理学院,武汉430065 [3]冶金工业过程系统科学湖北省重点实验室,武汉430065

出　　处：《数学学报(中文版)》2025年第2期304-324,共21页Acta Mathematica Sinica:Chinese Series

基　　金：国家自然科学基金(12171378);福建省自然科学基金项目(2023J011101);福建省教育厅中青年教师教育科研项目(JAT210360);福建江夏学院科研培育人才基金资助项目(JXZ2022012)。

摘　　要：在R^(n)中,利用凸体的弦幂积分及其不等式,得到若干关于μ-随机弦长、ν-随机弦长、λ-随机弦长的k-阶矩之间的不等式.根据凸体的弦幂积分与包含函数的关系,得到以上三种随机弦长矩的一种新的表达形式.利用μ-随机弦长的分布函数及其概率密度函数的性质,分别得到ν-随机弦长的分布函数及其概率密度函数、λ-随机弦长的分布函数及其概率密度函数的计算公式,并建立三种分布函数之间的关系.在此基础上,以R2中的菱形域、正五边形域、正六边形域为例,分别得到相应的三种随机弦长的一阶矩以及ν-随机弦长的分布函数的具体结果.Using the chord power integral and its inequalities of a convex body,we establish inequalities about moments forμ-random chord length,v-random chord length,and>-random chord length in R^(n).Based on the relationship between the chord power integral and containment function of a convex body,we obtain a new expression for moments of three kinds of random chord length mentioned above.By utilizing the properties of the distribution function and probability density function ofμ-random chord length,we get the calculation formulas for the distribution function and probability density function of v-random chord length,and the distribution function and probability density function of>-random chord length,respectively.Further,we establish the relationships among three kinds of distribution functions.On this basis,taking a rhombus,regular pentagon,and regular hexagon as examples in R^(2),we give the expressions of their 1-order moment for three kinds of random chord length and the distribution function of v-random chord length.

关 键 词：随机弦长矩 分布函数 弦幂积分 包含函数 概率密度 

分 类 号：O186.5[理学—数学]