Abstract:

Based on the relationship between the average curvature integrals of the convex and flat convex bodies in ${{\mathbb{R}}^{n}}$, the geometric probability that $h$ linear subspaces intersecting the convex body $K$ within $K$ is given. On this basis, the probability of the existence of a sphere with a radius of $r$ and a common point with all random planes in the convex body $K$ is given. Furthermore, the convergence probabilities of the points in the one-dimensional and two-dimensional cases are discussed.

Key words: mean curvature integral, parallel convex sets, geometric probability, homogeneous integration, minkowski addition