Abstract:

The probability that three independent random subspaces in $\mathbb{R}^n$ intersecting a convex body K have their common point intersecting K is found by using of the mean curvature integral of convex sets. Then we focus on the particular case of hyperplanes. On the base of this, we state the geometric probability of hyperplanes that intersect a ball, a cube or a right parallelepiped having an intersection inside the same ball, the cube or the right parallelepiped respectively. Finally, the monotonicity, convergence, and size relationship of the geometric probabilistic sequence are discussed.

Key words: Mean curvature integral, Geometric probability, Buffon needle throwing, Elementary symmetric function, Hyperplanes, Subspaces