Abstract
In this paper, we prove that for any seven points in a unit square there exist three points whose area is not greater than a constant h 7 = 0.083859... as conjectured by Francesc Comellas and J. Luis A. Yebra in 2002.
This work is supported by the National Natural Science Foundation of China (No. 10471044) and the Major Research Plan of the National Natural Science Foundation of China (No. 90718041).
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References
Aichholzer, O., Aurenhammer, F., Krasser, H.: Enumerating order types for small point sets with applications. Order 19(3), 265–281 (2002)
Aichholzer, O., Krasser, H.: Abstract order type extension and new results on the rectilinear crossing number. Computational Geometry: Theory and Applications, Special Issue on the 21st European Workshop on Computational Geometry 36(1), 2–15 (2006)
Comellas, F., Yebra, J.L.A.: New lower bounds for heilbronn numbers. Electr. J. Comb. 9(6), 1–10 (2002)
Dress, A.W.M., 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 Sci. Publishing, Singapore (1995)
Goldberg, M.: Maximizing the smallest triangle made by n points in a square. Math. Magazine 45(3), 135–144 (1972)
Yang, L., Zeng, Z.: Heilbronn problem for seven points in a planar convex body. In: Dingzhu, D., Pardalos, P.M. (eds.) Minimax and Applications (1995)
Yang, L., Zhang, J., Zeng, Z.: On exact values of heilbronn numbers for triangular regions. Tech. Rep. 91-098, Universität Bielefeld (1991)
Yang, L., Zhang, J., Zeng, Z.: On goldberg’s conjecture: Computing the first several heilbronn numbers. Tech. Rep. 91-074, Universität Bielefeld (1991)
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)
Zeng, Z., Shan, M.: Semi-mechanization method for an unsolved optimization problem in combinatorial geometry. In: Proceedings of the 2007 ACM Symposium on Applied Computing, pp. 762–766. ACM, New York (2007)
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Zeng, Z., Chen, L. (2011). On the Heilbronn Optimal Configuration of Seven Points in the Square. In: Sturm, T., Zengler, C. (eds) Automated Deduction in Geometry. ADG 2008. Lecture Notes in Computer Science(), vol 6301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21046-4_11
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DOI: https://doi.org/10.1007/978-3-642-21046-4_11
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