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Zapiski Nauchnykh Seminarov POMI, 2023, Volume 525, Pages 134–149
(Mi znsl7373)
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On the average area of a triangle inscribed in a convex figure
A. S. Tokmachev Saint Petersburg State University
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
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
Linking options:
https://www.mathnet.ru/eng/znsl7373 https://www.mathnet.ru/eng/znsl/v525/p134
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Statistics & downloads: |
Abstract page: | 47 | Full-text PDF : | 32 | References: | 19 |
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