Random sections of spherical convex bodies

TL;DR

This paper derives the distribution of distances between random points in convex spherical bodies, explicitly relating it to the distribution of random chord lengths, with specific results for spherical caps.

Contribution

It provides explicit formulas connecting the distributions of point distances and chord lengths in convex spherical bodies, including spherical caps.

Findings

Distribution of point distances expressed via chord length distribution

Explicit density formula for spherical caps

Analytical relationship between geometric measures

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.