Geodesic
Random sections of convex bodies
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 28, Tome 486 (2019), pp. 190-199

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Consider a convex body $D$ in $\mathbb{R}^n$. We obtain an explicit formula expressing the distribution function of the distance between two random points uniformly and independently chosen in $D$ in terms of the distribution function of the length of a random chord of $D$. As a corollary, we derive Kingman's formula which connects the moments of these distributions. 
@article{ZNSL_2019_486_a10,
     author = {T. Moseeva},
     title = {Random sections of convex bodies},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {190--199},
     publisher = {mathdoc},
     volume = {486},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_486_a10/}
}
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T. Moseeva. Random sections of convex bodies. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 28, Tome 486 (2019), pp. 190-199. http://geodesic.mathdoc.fr/item/ZNSL_2019_486_a10/