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An algorithm for Heilbronn's problem

Session 2: Computational Geometry

Part of the Lecture Notes in Computer Science book series (LNCS,volume 1276)

Abstract

Heilbronn conjectured that given arbitrary n points from R 2, located in the unit square (or disc), there must be three points which form a triangle of area at most O(1/n2). This conjecture was shown to be false by a nonconstructive argument of Komlós, Pintz and Szemerédi [6] who showed that for every n there is a configuration of n points in the unit square where all triangles have area at least Ω(log n/n2). In this paper, we provide a polynomial-time algorithm which for every n computes such a configuration of n points.

We then consider a generalization of this problem as introduced by Schmidt [10] to convex hulls of k points. We obtain the following result: For every k ≥ 4, there is a polynomial-time algorithm which on input n computes n points in the unit square such that the convex hull of any k points has area at least Ω(1/n(k-i)/(k-2)). For k = 4, the existence of such a configuration has been proved in [10].

Keywords

  • Grid Point
  • Convex Hull
  • Minimum Area
  • London Mathematical Society
  • Simple Geometric Argument

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This research was supported by the Deutsche Forschungsgemeinschaft as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).

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References

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© 1997 Springer-Verlag Berlin Heidelberg

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Bertram-Kretzberg, C., Hofmeister, T., Lefmann, H. (1997). An algorithm for Heilbronn's problem. In: Jiang, T., Lee, D.T. (eds) Computing and Combinatorics. COCOON 1997. Lecture Notes in Computer Science, vol 1276. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0045069

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  • DOI: https://doi.org/10.1007/BFb0045069

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  • Print ISBN: 978-3-540-63357-0

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