JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (4): 76-85.doi: 10.6040/j.issn.1671-9352.0.2020.356

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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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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

CLC Number: 

  • O186.5
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