A note on diameters of point sets

Abstract

Relationships between the diameter of a set of n points in the plane at mutual distance at least one, the diameter of an equilateral n-gon and the radius of a circle including n unit disks are explored. Upper bounds on the minimal diameter of a point set at mutual distance at least one are presented for up to 30 points.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. 1

    Abramson M.A., Audet C., Dennis J.E. Jr, Le Digabel S.: Orthomads: a deterministic mads instance with orthogonal directions. SIAM J. Optim. 20(2), 948–966 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  2. 2

    Audet C., Béchard V., Le Digabel S.: Nonsmooth optimization through mesh adaptive direct search and variable neighborhood search. J. Global Optim. 41(2), 299–318 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3

    Audet C., Hansen P., Jaumard B., Savard G.: A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Math. Program. 87(1, Ser. A), 131–152 (2000)

    MATH  MathSciNet  Google Scholar 

  4. 4

    Audet, C., Hansen, P., Messine, F.: http://messine.perso.enseeiht.fr/DPS/

  5. 5

    Audet C., Hansen P., Messine F.: Extremal problems for convex polygons. J. Glob. Optim. 38(2), 163–179 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6

    Audet C., Hansen P., Messine F.: Extremal problems for convex polygons—an update. In: Pardalos, P.M., Coleman, T.F. (eds) Lectures on global optimization. Fields Institute Communications, vol. 55, American Mathematical Society, New York (2009)

    Google Scholar 

  7. 7

    Audet C., Hansen P., Messine F., Perron S.: The minimum diameter octagon with unit-length sides: Vincze’s wife’s octagon is suboptimal. J. Combin. Theory Ser. A 108(1), 63–75 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  8. 8

    Bateman P., Erdős P.: Geometrical extrema suggested by a lemma of Besicovitch. Am. Math. Mon. 58, 306–314 (1951)

    MATH  Article  Google Scholar 

  9. 9

    Bezdek A., Fodor F.: Minimal diameter of certain sets in the plane. J. Comb. Theory Ser. A 85, 105–111 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  10. 10

    Cantrell, D.: Minimizing the ratio of maximum to minimum distance. http://www2.stetson.edu/~efriedma/maxmin/

  11. 11

    Croft H.T., Falconer K.J., Guy R.K.: Unsolved Problems in Geometry. Springer, New York (1991)

    Google Scholar 

  12. 12

    Melissen, H.: Packing and Covering with Circles. Ph.D. thesis, Universiteit Utrecht, The Netherlands (1997)

  13. 13

    Mossinghoff M.J.: An isodiametric problem for equilateral polygons. In: Tapasinexperimentalmathematics. Amdeberham, T., Moll, V. (eds) Contemporary Mathematics, vol 457, pp. 237–252. American Mathematical Society, New York (2008)

    Google Scholar 

  14. 14

    Reinhardt K.: Extremale polygone gegebenen durchmessers. Jahresber. Deutsch. Math. Verein 31, 251–270 (1922)

    Google Scholar 

  15. 15

    Reuleaux F.: The Kinematics of Machinery. Dover, New York (1963)

    Google Scholar 

  16. 16

    Specht, E.: Circles in a circle. http://www.packomania.com

  17. 17

    Tamvakis N.K.: On the perimeter and the area of the convex polygon of a given diameter. Bull. Greek Math. Soc. 28, 115–132 (1987)

    MATH  MathSciNet  Google Scholar 

  18. 18

    Vincze S.: On a geometrical extremum problem. Acta Sci. Math. Szeged. 12, 136–142 (1950)

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Charles Audet.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Audet, C., Fournier, X., Hansen, P. et al. A note on diameters of point sets. Optim Lett 4, 585–595 (2010). https://doi.org/10.1007/s11590-010-0185-y

Download citation

Keywords

  • Point set
  • Diameter
  • Circle packing
  • Disk packing