Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

In the paper, a formula to calculate the probability that a random segment L(ω, u) in R n with a fixed direction u and length l lies entirely in the bounded convex body D ⊂ R n ( 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.