Discrete & Computational Geometry

, Volume 49, Issue 3, pp 589–600 | Cite as

The Small Octagons of Maximal Width

  • Charles Audet
  • Pierre Hansen
  • Frédéric Messine
  • Jordan Ninin


The paper answers an open problem introduced by Bezdek and Fodor (Arch. Math. 74:75–80, 2000). The width of any unit-diameter octagon is shown to be less than or equal to \(\frac{1}{4}\sqrt{10 + 2\sqrt{7}}\) and there are infinitely many small octagons having this optimal width. The proof combines geometric and analytical reasoning as well as the use of a recent version of the deterministic and reliable global optimization code IBBA based on interval and affine arithmetics. The code guarantees a certified numerical accuracy of \(1\times 10^{-7}\).


Polygon Octagon Width Diameter 


  1. 1.
    Audet, C., Ninin, J.: Maximal perimeter, diameter and area of equilateral unit-width convex polygons. J. Glob. Optim. (2011). doi:10.1007/s10898-011-9780-4
  2. 2.
    Audet, C., Hansen, P., Messine, F., Xiong, J.: The largest small octagon. J. Combin. Theory Ser. A 98(1), 46–59 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Audet, C., Hansen, P., Messine, F.: Extremal problems for convex polygons. J. Glob. Optim. 38(2), 163–179 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Audet, C., Hansen, P., Messine, F.: The small octagon with longest perimeter. J. Comb. Theory Appl. Ser. A 114(1), 135–150 (2007)MathSciNetMATHCrossRefGoogle 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, pp. 1–16. American Mathematical Society, Providence, RI (2009).Google Scholar
  7. 7.
    Audet, C., Hansen, P., Messine, F.: Isoperimetric polygons of maximum width. Discrete Comput. Geom. 41(1), 45–60 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bezdek, A., Fodor, F.: On convex polygons of maximal width. Arch. Math. 74, 75–80 (2000)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    de Figueiredo, L., Stolfi, J.: Affine arithmetic: concepts and applications. Numer. Algorithms 37(1–4), 147–158 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gashkov, S.: Inequalities for the area and perimeter of a convex polygon. Kwant 10, 15–19 (1985). In Russian: Open image in new window Open image in new window 10, 15–19 (1985).Google Scholar
  11. 11.
    Graham, R.L.: The largest small hexagon. J. Combin. Theory 18, 165–170 (1975)MATHCrossRefGoogle Scholar
  12. 12.
    Messine, F.: Extensions of affine arithmetic: application to unconstrained global optimization. J. Univ. Comput. Sci. 8(11), 992–1015 (2002)MathSciNetGoogle Scholar
  13. 13.
    Messine, F.: Deterministic global optimization using interval constraint propagation techniques. RAIRO Oper. Res. 38(4), 277–293 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Messine, F., Touhami, A.: A general reliable quadratic form: an extension of affine arithmetic. Reliab. Comput. 12(3), 171–192 (2006)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Moore, R.E.: Interval Analysis. Prentice-Hall Inc., Englewood Cliffs, NJ (1966)MATHGoogle Scholar
  16. 16.
    Mossinghoff, M.J.: Enumerating isodiametric and isoperimetric polygons. J. Comb. Theory Ser. A 118(6), 1801–1815 (2011)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer linear programming. Math. Program. Ser. A 99(2), 283–296 (2004)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Ninin, J.: Optimisation globale basée sur l’analyse d’intervalles: relaxation affine et limitation de la mémoire. PhD thesis, Institut National Polytechnique de Toulouse (2010). http://ethesis.inp-toulouse.fr/archive/00001477/01/Ninin.pdf
  19. 19.
    Ninin, J., Messine, F.: A metaheuristic methodology based on the limitation of the memory of interval branch and bound algorithms. J. Glob. Optim. 50, 629–644 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. Technical Report G-2010-31, GERAD (2010). http://www.optimization-online.org/DB_HTML/2012/10/3650.html
  21. 21.
    Pál, J.: Ein Minimumproblem für Ovale. Math. Ann. 83, 311–319 (1921)Google Scholar
  22. 22.
    Reuleaux, F.: The Kinematics of Machinery. Dover, New York, NY (1963)Google Scholar
  23. 23.
    Revol, N.: Standardized interval arithmetic and interval arithmetic used in libraries. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) Mathematical Software-ICMS 2010, vol. 6327, pp. 337–341. Springer, Berlin (2010) WOS:000286346100054Google Scholar
  24. 24.
    Yaglom, I.M., Boltyanskiĭ, V.G.: Convex Figures (Translated by P.J. Kelly, L.F. Walton, Lib. Math. Circle, vol. 4) Rinehart and Winston, New York, NY (1961)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Charles Audet
    • 1
  • Pierre Hansen
    • 2
  • Frédéric Messine
    • 3
  • Jordan Ninin
    • 4
  1. 1.GERAD and Département de Mathématiques et de Génie IndustrielÉcole Polytechnique de MontréalMontrealCanada
  2. 2.GERAD and Département des Méthodes QuantitativesHEC MontréalMontrealCanada
  3. 3.ENSEEIHT-IRITUniversity of ToulouseToulouse Cedex 7France
  4. 4.IHSEV Team, LAB-STICC, UMR-CNRS 3162ENSTA-BretagneBrestFrance

Personalised recommendations