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Zapiski Nauchnykh Seminarov POMI, 2023, Volume 525, Pages 134–149 (Mi znsl7373)  

On the average area of a triangle inscribed in a convex figure

A. S. Tokmachev

Saint Petersburg State University
References:
Abstract: 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 $K$.
Key words and phrases: geometric inequalities, Blaschke's inequality, integral geometry, Hausdorff metric, Fourier series, mean area.
Received: 17.10.2023
Document Type: Article
Language: Russian
Citation: A. S. Tokmachev, “On the average area of a triangle inscribed in a convex figure”, Probability and statistics. Part 34, Zap. Nauchn. Sem. POMI, 525, POMI, St. Petersburg, 2023, 134–149
Citation in format AMSBIB
\Bibitem{Tok23}
\by A.~S.~Tokmachev
\paper On the average area of a triangle inscribed in a convex figure
\inbook Probability and statistics. Part~34
\serial Zap. Nauchn. Sem. POMI
\yr 2023
\vol 525
\pages 134--149
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl7373}
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  • https://www.mathnet.ru/eng/znsl/v525/p134
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