Random sections of spherical convex bodies

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 $\Delta (K)$ via the distribution of $\sigma (K)$. From this we find the density of distribution of $\Delta (K)$ when $K$ is a spherical cap.