Geodesic
Mean distance between random points on the boundary of a convex body
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 32, Tome 510 (2022), pp. 248-261
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Consider a convex figure $K$ on the plane. Let $\theta(K)$ denote the mean distance between two random points independently and uniformly selected on the boundary of $K$. The main result of the paper is that among all convex shapes of a fixed perimeter, the maximum value of $\theta(K)$ is reached at the circle and only at it. The continuity of $\theta(K)$ in the Hausdorff metric is also proved. 
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A. S. Tokmachev. Mean distance between random points on the boundary of a convex body. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 32, Tome 510 (2022), pp. 248-261. http://geodesic.mathdoc.fr/item/ZNSL_2022_510_a14/

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