Geodesic
Random sections of spherical convex bodies
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 198-208

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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. 
@article{ZNSL_2020_495_a11,
     author = {T. D. Moseeva and A. S. Tarasov and D. N. Zaporozhets},
     title = {Random sections of spherical convex bodies},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {198--208},
     publisher = {mathdoc},
     volume = {495},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a11/}
}
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T. D. Moseeva; A. S. Tarasov; D. N. Zaporozhets. Random sections of spherical convex bodies. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 198-208. http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a11/