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International Workshop on Computer Algebra in Scientific Computing

CASC 2019: Computer Algebra in Scientific Computing pp 458–477Cite as

Determining the Heilbronn Configuration of Seven Points in Triangles via Symbolic Computation

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

Abstract

In this paper we first recall some rigorously proved results related to the Heilbronn numbers and the corresponding optimal configurations of \(n=5,6,7\) points in squares, disks, and general convex bodies K in the plane, \(n=5,6\) points in triangles and a bundle of approximate results obtained by numeric computation in the Introduction section. And then in the second section we will present a proof to a conjecture on the Heilbronn number for seven points in the triangle through solving a group of non-linear optimization problems via symbolic computation. In the third section we list three unsolved well-formed such non-linear programming problems corresponding to Heilbronn configurations for \(n=8,9\) points in squares and 8 points in triangle, we expect they can be solved by similar method we used in the Section two. In the final section we mention two generalizations of the classic Heilbronn triangle problem. The paper aims to provide a concise guide to further studies on Heilbronn-type problems for small number of points in specific convex bodies.

Supported by the grant from the National Natural Science Foundation of China (No. 11471209).

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References

  1. Barequet, G.: A lower bound for for Heilbronn’s triangle problem in \(d\) dimensions. SIAM J. Discrete Math. 14(2), 230–236 (2001)

    Article  MathSciNet  Google Scholar 

  2. Barequet, G., Shaikhet, A.: The on-line Heilbronn’s triangle problem in \(d\) dimensions. Discrete Comput. Geom. 38(1), 51–60 (2007)

    Article  MathSciNet  Google Scholar 

  3. Brass, P.: An upper bound for the \(d\)-dimensional analogue of Heilbronn’s triangle problem. SIAM J. Discrete Math. 19(1), 192–195 (2006)

    Article  MathSciNet  Google Scholar 

  4. Cantrell, D.: The Heilbronn problem for triangles. http://www2.stetson.edu/efriedma/heiltri/. Accessed 5 Apr 2019

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. Comellas, F., Yebra, J.L.A.: New lower bounds for Heilbronn numbers. Elec. J. Comb. 9, 10 (2002). http://www.combinatorics.org/Volume_9/PDF/v9i1r6.pdf. Accessed 5 Apr 2019

  8. De Comité, F., Delahaye, J.-P.: Automated proofs in geometry : computing upper bounds for the Heilbronn problem for triangles. https://arxiv.org/pdf/0911.4375.pdf. Accessed 5 Apr 2019

  9. Dress, A., Yang, L., Zeng, Z.: Heilbronn problem for six points in a planar convex body. Combinatorics and Graph Theory 1995. vol. 1 (Hefei), pp. 97–118, World Sci. Publishing, River Edge (1995)

    Google Scholar 

  10. Dress, A., Yang, L., Zeng, Z.: Heilbronn problem for six 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. 173–190. Springer, Boston (1995)

    Chapter  Google Scholar 

  11. Friedman, E.: The Heilbronn problem for squares. https://www2.stetson.edu/friedma/heilbronn/. Accessed 5 Apr 2019

  12. Friedman, E.: The Heilbronn problem for triangles. https://www2.stetson.edu/efriedma/heiltri/. Accessed 5 Apr 2019

  13. Gavin, H., Scruggs, J.: Constrained optimization using Lagrange multipliers. http://people.duke.edu/hpgavin/cee201/LagrangeMultipliers.pdf

  14. Goldberg, M.: Maximizing the smallest triangle made by points in a square. Math. Mag. 45, 135–144 (1972)

    Article  MathSciNet  Google Scholar 

  15. Kahle, M.: Points in a triangle forcing small triangles. Geombinatorics XVIII, 114–128 (2008). arxiv.org/abs/0811.2449v1

    MathSciNet  Google Scholar 

  16. Karpov, P.: Notable results. https://inversed.ru/Ascension.htm#results. Accessed 5 Apr 2019

  17. Kómlos, J., Pintz, J., Szemerédi, E.: On Heilbronn’s triangle problem. J. London Math. Soc. 24, 385–396 (1981)

    Article  MathSciNet  Google Scholar 

  18. Kómlos, J., Pintz, J., Szemerédi, E.: A lower bound for Heilbronn’s triangle problem. J. London Math. Soc. 25, 13–24 (1982)

    Article  MathSciNet  Google Scholar 

  19. Pegg Jr., E.: Heilbronn triangles in the unit square. Wolfram Demonstrations Project. http://demonstrations.wolfram.com/HeilbronnTrianglesintheunitSquare/. Accessed 6 Apr 2019

  20. Roth, K.F.: On a problem of Heilbronn. J. London Math. Soc. 26, 198–204 (1951)

    Article  MathSciNet  Google Scholar 

  21. Roth, K.F.: On a problem of Heilbronn II. Proc. London Math Soc. 3(25), 193–212 (1972)

    Article  MathSciNet  Google Scholar 

  22. Roth, K.F.: On a problem of Heilbronn III. Proc. London Math Soc. 3(25), 543–549 (1972)

    Article  MathSciNet  Google Scholar 

  23. Roth, K.F.: Estimation of the area of the smallest triangle obtained by selecting three out of \(n\) points in a disc of unit area. Proc. Symp. Pure Mathematics 24, AMS, Providence, 251–262 (1973)

    Google Scholar 

  24. Roth, K.F.: Developments in Heilbronn’s triangle problem. Adv. Math. 22, 364–385 (1976)

    Article  MathSciNet  Google Scholar 

  25. Schmidt, W.: On a problem of Heilbronn. J. London Math. Soc. 4, 545–550 (1971/1972)

    Article  MathSciNet  Google Scholar 

  26. Sturmfels, S: Solving systems of polynomial equations. https://math.berkeley.edu/bernd/cbms.pdf. Accessed 25 May 2019

  27. Tal, A.: Algorithms for Heilbronn’s triangle problem. Israel Institute of Technology. Master Thesis. May 2009

    Google Scholar 

  28. Weisstein E.W.: Heilbronn triangle problem. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/HeilbronnTriangleProblem.html

  29. 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)

    Chapter  Google Scholar 

  30. Yang, L., Zhang, J., Zeng, Z.: Heilbronn problem for five points. Int’l Centre Theoret. Physics preprint IC/91/252 (1991)

    Google Scholar 

  31. Yang, L., Zhang, J., Zeng, Z.: On Goldberg’s conjecture: Computing the first several Heilbronn numbers. Universität Bielefeld, Preprint 91–074 (1991)

    Google Scholar 

  32. Yang, L., Zhang, J., Zeng, Z.: On exact values of Heilbronn numbers for triangular regions. Universität Bielefeld, Preprint 91–098 (1991)

    Google Scholar 

  33. Yang, L., Zhang, J., Zeng, Z.: A conjecture on the first several Heilbronn numbers and a computation. Chinese Ann. Math. Ser. A 13, 503–515 (1992)

    MathSciNet  MATH  Google Scholar 

  34. Yang, L., Zhang, J., Zeng, Z.: On the Heilbronn numbers of triangular regions. Acta Math Sinica 37, 678–689 (1994)

    MathSciNet  MATH  Google Scholar 

  35. 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_11

    Chapter  Google Scholar 

  36. Zeng, Z., Chen, L.: Heilbronn configuration of eight points in the square. Manuscript. 1–74

    Google Scholar 

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Author information

Authors and Affiliations

  1. Shanghai University, 99 Shangda Road, Shanghai, 200444, China

    Zhenbing Zeng

  2. East China Normal University, 3633 North Zhongshan Road, Shanghai, 200062, China

    Liangyu Chen

Authors
  1. Zhenbing Zeng
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  2. Liangyu Chen
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Corresponding authors

Correspondence to Zhenbing Zeng or Liangyu Chen .

Editor information

Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Zeng, Z., Chen, L. (2019). Determining the Heilbronn Configuration of Seven Points in Triangles via Symbolic Computation. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_30

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  • DOI: https://doi.org/10.1007/978-3-030-26831-2_30

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26830-5

  • Online ISBN: 978-3-030-26831-2

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