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A Heilbronn Type Inequality for Plane Nonagons

  • Zhenbing Zeng
  • Jian LuEmail author
  • Lydia DehbiEmail author
  • Liangyu Chen
  • Jianlin WangEmail author
Conference paper
  • 61 Downloads
Part of the Communications in Computer and Information Science book series (CCIS, volume 1125)

Abstract

In this paper, we present a proof of the property that for any convex nonagon \(P_1P_2\ldots P_9\) in the plane, the smallest area of a triangle \(P_{i}P_{j}P_{k} (1\le i< j < k \le 9)\) is at most a fraction of \(4\cdot \sin ^2(\pi /9)/9= 0.05199\ldots \) of the area of the nonagon. The problems is transformed into an optimization problem with bilinear constraints and solved by symbolic computation with Maple.

Keywords

Heilbronn problem Convex nonagon Computer algebra Lagrange multipliers 

References

  1. 1.
    Behrend, F.: Über die kleinste umbeschriebene und die größte einbeschriebene Ellipse eines konvexen. Bereichs. Math. Ann. 115(1), 379–411 (1938)CrossRefGoogle Scholar
  2. 2.
    Buitrago, A., Huylebrouck, D.: Nonagons in the Hagia Sophia and the Selimiye Mosque. Nexus Netw. J. 17(1), 157–181 (2015)CrossRefGoogle Scholar
  3. 3.
    Cantrell, D.: The Heilbronn problem for triangles. http://www2.stetson.edu/~efriedma/heiltri/. Accessed 7 Sept 2019
  4. 4.
    Chen, L., Zeng, Z., Zhou, W.: An upper bound of Heilbronn number for eight points in triangles. J. Comb. Optim. 28(4), 854–874 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, L., Xu, Y., Zeng, Z.: Searching approximate global optimal Heilbronn configurations of nine points in the unit square via GPGPU computing. J. Global Optim. 68(1), 147–167 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Comellas, F., Yebra, J.L.A.: New lower bounds for Heilbronn numbers. Electron. J. Comb. 9, #R6 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dress, A., Yang, L., Zeng Z.: Heilbronn problem for six points in a planar convex body. In: Combinatorics and Graph Theory 1995, vol. 1 (Hefei), pp. 97–118. World Scientific Publishing, River Edge (1995)Google Scholar
  8. 8.
    Friedman, E.: The Heilbronn Problem. http://www.stetson.edu/efriedma/heilbronn. Accessed 14 Mar 2019
  9. 9.
    Goldberg, M.: Maximizing the smallest triangle made by \(n\) points in a square. Math. Mag. 45, 135–144 (1972)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Komlós, J., Pintz, J., Szemerédi, E.: On Heilbronn’s triangle problem. J. London Math. Soc. 24(3), 385–396 (1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Komlós, J., Pintz, J., Szemerédi, E.: A lower bound for Heilbronn’s problem. J. London Math. Soc. 25(1), 13–24 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Weisstein, E.W.: Heilbronn Triangle Problem. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/HeilbronnTriangleProblem.html. Accessed 14 Mar 2019
  13. 13.
    Yang, L., Zeng, Z.: Heilbronn problem for seven points in a planar convex body. In: Du, D.-Z., Pardalos, P.M. (eds.) Minimax and Applications. Nonconvex Optimization and Its Applications, vol. 4, pp. 191–218. Springer, Boston (1995).  https://doi.org/10.1007/978-1-4613-3557-3_14CrossRefGoogle Scholar
  14. 14.
    Yang, L., Zhang, J., Zeng, Z.: Heilbronn problem for five points. Intl Centre Theoret. Physics preprint IC/91/252 (1991)Google Scholar
  15. 15.
    Yang, L., Zhang, J., Zeng, Z.: A conjecture on the first several Heilbronn numbers and a computation. Chinese Ann. Math. Ser. A 13(2), 503–515 (1992). (in Chinese)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Yang, L., Zhang, J., Zeng, Z.: On the Heilbronn numbers of triangular regions. Acta Math. Sinica 37(5), 678–689 (1994). (in Chinese)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zeng, Z., Chen, L.: On the Heilbronn optimal configuration of seven points in the square. In: Sturm, T., Zengler, C. (eds.) ADG 2008. LNCS (LNAI), vol. 6301, pp. 196–224. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-21046-4_11CrossRefGoogle Scholar
  18. 18.
    Zeng, Z., Chen, L.: Determining the Heilbronn configuration of seven points in triangles via symbolic computation. In: England, M., Koepf, W., Sadykov, T.M., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2019. LNCS, vol. 11661, pp. 458–477. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-26831-2_30CrossRefGoogle Scholar
  19. 19.
    Zeng, Z., Shan, M.: Semi-mechanization method for an unsolved optimization problem in combinatorial geometry. In: SAC 2007, pp. 762–766 (2007)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.East China Normal UniversityShanghaiChina
  3. 3.Henan UniversityHenanChina

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