Abstract:
Let $K\subset\mathbb S^{d-1}$ be a convex spherical body. Denote by $\Delta(K)$ the distance between two random points in $K$ and denote by $\sigma(K)$ the length of a random chord of $K$. We explicitly express the distribution of $\Delta(K)$ viathe distribution of $\sigma(K)$. From this we find the density of distribution of $\Delta(K)$ when $K$ is a spherical cap.
Key words and phrases:Crofton formula, mean distance, spherical Blaschke–Petkantschin formula, spherical integral geometry, spherical convex body, random chord.