Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

Abstract

In the paper, a formula to calculate the probability that a random segment L(ω, u) in Rⁿ with a fixed direction u and length l lies entirely in the bounded convex body D ⊂ Rⁿ (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ⊂ D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ⊂ D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.