On the average perimeter of the inscribed random polygon

Authors

  • Yakov Yu. Nikitin Saint Petersburg State University; National Research University Higher School of Economics https://orcid.org/0000-0002-4513-5713
  • Tatiana A. Polevaya St. Petersburg National Research University of Information Technologies, Mechanics and Optics

DOI:

https://doi.org/10.21638/11701/spbu01.2020.108

Abstract

Suppose we put on the unit circumference n independent uniformly distributed random points and build a convex random polygon with the vertices in these points. What are the average area and the average perimeter of this polygon? The average area was calculated by K. Brown some years ago. We calculatе the average perimeter and obtain quite similar formulae. In the same time we discuss the rate of convergence of this value to the limit. We evaluate also the average value of the sum of squares for the sides of the inscribed triangle.

Keywords:

random polygon, perimeter, convexity, uniform distribution

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References

Литература

Yaglom I. M., Boltyanskii V. G. Convex figures. Holt, Rinehart and Winston, 1961.

Lao W., Mayer M. U-max-statistics // Journ. Multiv. Anal. 2008. Vol. 99. P. 2039–2052.

Mayer M. Random Diameters and Other U-max-Statistics. Ph.D. Thesis, Bern University, 2008.

Koroleva E. V., Nikitin Ya. Yu. U-max-statistics and limit theorems for perimeters and areas of random polygons // Journ. Multiv. Anal. 2014. Vol. 127, N 5. P. 98–111.

Brown K. Expected Area of Random Polygon In a Circle. Available at: https://www.mathpages.com/home/kmath516/kmath516.htm (accessed: September 19, 2019).

Gradshteyn I. S., Ryzhik I. M. Table of integrals, series and products. Academic Press: 1980.

Прасолов В. В. Задачи по планиметрии. М.: МЦНМО, 2001.

Алексеев В. М., Галеев Э. М., Тихомиров В. М. Сборник задач по оптимизации. Теория. Примеры. Задачи. M.: Наука, 1984.

Ioffe A., Tikhomirov V. Extremal Problems. Amsterdam: North Holland, 1979

References

Yaglom I. M., Boltyanskii V. G., Convex figures (Holt, Rinehart and Winston, 1961).

Lao W., Mayer M., “U-max-statistics”, Journ. Multiv. Anal. 99, 2039–2052 (2008).

Mayer M., Random Diameters and Other U-max-Statistics (Ph.D. Thesis, Bern University, 2008).

Koroleva E. V., Nikitin Ya. Yu., “U-max-statistics and limit theorems for perimeters and areas of random polygons”, Journ. Multiv. Anal. 127(5), 98–111 (2014).

Brown K., Expected Area of Random Polygon In a Circle. Available at: https://www.mathpages.com/home/kmath516/kmath516.htm (accessed: September 19, 2019).

Gradshteyn I. S., Ryzhik I. M., Table of integrals, series and products (Academic Press, 1980).

Prasolov V., Problems in planimetry (MTsNMO Publ., Мoscow, 2001). (In Russian)

Alekseev V. M., Galeev E. M., Tikhomirov V. M., Collection of problems on optimization. Theory. Examples. Problems (Nauka Publ., Moscow, 1984). (In Russian)

Ioffe A., Tikhomirov V., Extremal Problems (North Holland Publ., Amsterdam, 1979).

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Published

2020-05-13

How to Cite

Nikitin, Y. Y., & Polevaya, T. A. (2020). On the average perimeter of the inscribed random polygon. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(1), 77–84. https://doi.org/10.21638/11701/spbu01.2020.108

Issue

Vol. 7 No. 1 (2020)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.