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《山东大学学报(理学版)》 ›› 2021, Vol. 56 ›› Issue (4): 76-85.doi: 10.6040/j.issn.1671-9352.0.2020.356

• • 上一篇    

En中的几何概率及其极值

赵江甫   

  1. 福建江夏学院数理教研部, 福建 福州 350108
  • 发布日期:2021-04-13
  • 作者简介:赵江甫(1985— ),女,硕士,讲师,研究方向为积分几何与几何概率. E-mail:2833811196@qq.com
  • 基金资助:
    福建省教育厅中青年教师教育科研基金资助项目(JT180585);福建江夏学院科研培育人才基金资助项目(JXZ2019016)

Geometric probability and its extremes in En

ZHAO Jiang-fu   

  1. Department of Mathematics and Physics, Fujian Jiangxia University, Fuzhou 350108, Fujian, China
  • Published:2021-04-13

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摘要： 三维空间中,三个与凸体K相交的平面的公共点落入K内的概率已有结果。为了将此结论推广到更一般的n维欧式空间,设L、G、H为En中与凸体K相交的3个超平面,利用积分几何的方法,给出超平面束的交L∩G∩H与凸体K相交的几何概率,并利用等周不等式,得到此概率序列的极大值。利用Minkowski不等式和Cauchy公式,给出En中超平面偶的交L∩G与凸体K相交的几何概率的极大值。根据上述结论,得到两个关于超几何函数的不等式。

关键词: 平均曲率积分, 几何概率, 蒲丰投针, 极值问题, 超平面束

Abstract: The probability that three planes in 3-dimensional Euclidean space intersecting a convex body K have their common point inside K has been obtained. In order to extend the above conclusion to more general n-dimensional Euclidean space, let L, G, H be three randomly chosen hyperplanes, that intersect a convex body K in En. The probability that L∩G∩H intersecting the convex body K is given by method of integral geometry. And then the Extreme value of this probabilistic sequence is obtained through isoperimetric inequalities. The maximum of probability that L∩G intersecting the convex body K is found by using of Minkowski inequality and Cauchys formula. As their application, two inequalities about hypergeometric functions are given.

Key words: mean curvature integral, geometric probability, Buffon needle throwing, extremum problem, hyperplanes

中图分类号: 

  • O186.5
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