Rn中超平面偶与特殊凸体相交的几何概率问题

利用凸体的均质积分给出了Rn中的超平面偶与n维正方体相交时，其交集也与此正方体相交的几何概率。此概率序列不仅与正方体棱长无关，而且关于维数n单调递增，并收敛于常数π/4。这一系列结果与n维球体时的情形类似。在此基础上，利用初等对称函数以及积分几何理论进一步讨论了棱长不等的n维长方体的情形，并给出了相应的几何概率的最大值。由于此几何概率序列与长方体的棱长有关，因此不再关于维数n单调递增，也不再具有收敛性，然而，当棱长满足一定条件时依然会收敛到常数π/4。

Geometric Probability of Pairs of Hyperplanes Intersecting withSpecial Bodies in Rn

This paper gives the geometric probability for pairs of hyperplanes which are intersecting with a n dimensional cube of equal lengths and the intersection are intersecting with the same cube in Rn by using the quermass integrale of a convex body. Then the monotonicity and convergence of the geometric probabilistic sequence are discussed. The geometric probabilistic sequence is not influenced by the change of lengths of the cube,and it converges monotonically increasing to the constant π/4,which is consistent with the case of sphere. On the base of this finding,this paper extends the related conclusions to the case of cuboid of unequal lengths by using elementary symmetric function and integral geometry theory and finds the maximum value. As the corresponding geometric probabilistic sequence is influenced by the change of lenths of the cuboid,it does not converge or monotonically increase with n any more. However,it still converges to the constant π/4 under certain conditions.