Skip to main content
SpringerLink
Log in

On the Average Perimeter of the Inscribed Random Polygon

  • MATHEMATICS
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

Assume that n independent uniformly distributed random points are set on the unit circle. Construct the convex random polygon with vertices in these points. What are the average area and the average perimeter of this polygon? Brown computed the average area several years ago. We compute the average perimeter and obtain quite similar expressions. We also discuss the rate of the convergence of this value to the limit and evaluate the average value of the sum of squares for the sides of the inscribed triangle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. I. M. Yaglom and V. G. Boltyanskii, Convex Figures (Holt, Rinehart and Winston, New York, 1961).

    MATH  Google Scholar 

  2. W. Lao and M. Mayer, “U-max-statistics,” J. Multivar. Anal. 99, 2039–2052 (2008).

    Article  MathSciNet  Google Scholar 

  3. M. Mayer, Random Diameters and Other U-max-Statistics, PhD Thesis (Univ. of Bern, Bern, 2008).

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

    Article  MathSciNet  Google Scholar 

  5. K. Brown, Expected Area of Random Polygon in a Circle. https://www.mathpages.com/home/kmath516/kmath516.htm. Accessed September 19, 2019.

  6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Nauka, Moscow, 1971; Academic, New York, 1980).

  7. V. Prasolov, Problems in Planimetry (MTsNMO, Moscow, 2001) [in Russian].

  8. V. M. Alekseev, E. M. Galeev, and V. M. Tikhomirov, Collection of Problems on Optimization. Theory. Examples. Problems (Nauka, Moscow, 1984) [in Russian].

    Google Scholar 

  9. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North Holland, Amsterdam, 1979).

Download references

Funding

The work of the first author is supported by St. Petersburg State University (grant SPbSU–DFG 6.65.37.2017); the work of the second author is supported by the Government of the Russian Federation (grant 08-08).

Author information

Authors and Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    Ya.Yu. Nikitin

  2. National Research University Higher School of Economics, 190008, St. Petersburg, Russia

    Ya.Yu. Nikitin

  3. St. Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101, St. Petersburg, Russia

    T. A. Polevaya

Authors
  1. Ya.Yu. Nikitin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. A. Polevaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ya.Yu. Nikitin or T. A. Polevaya.

Additional information

Translated by A. Muravnik

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nikitin, Y., Polevaya, T.A. On the Average Perimeter of the Inscribed Random Polygon. Vestnik St.Petersb. Univ.Math. 53, 58–63 (2020). https://doi.org/10.1134/S1063454120010070

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454120010070

Keywords:

Search

Navigation