Geodesic
On the average area of a triangle inscribed in a convex figure
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 34, Tome 525 (2023), pp. 134-149

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Let $K$ be a convex figure in the plane, and let $A, B, C$ be random points on its boundary given by a uniform distribution. In this paper, we prove that the maximum average area of triangle $ABC$ is obtained on the circle when the perimeter of $K$ is fixed. We also prove that the average area of the triangle is continuous in the Hausdorff metric as a functional of $K$. 
@article{ZNSL_2023_525_a10,
     author = {A. S. Tokmachev},
     title = {On the average area of a triangle inscribed in a convex figure},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {134--149},
     publisher = {mathdoc},
     volume = {525},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_525_a10/}
}
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A. S. Tokmachev. On the average area of a triangle inscribed in a convex figure. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 34, Tome 525 (2023), pp. 134-149. http://geodesic.mathdoc.fr/item/ZNSL_2023_525_a10/