Distinct Triangle Areas in a Planar Point Set

  • Adrian Dumitrescu
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4513)

Abstract

Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by n noncollinear points in the plane is at least \(\lfloor \frac{n-1}{2} \rfloor\), which is attained for ⌈n / 2⌉ and respectively \(\lfloor n/2\rfloor\) equally spaced points lying on two parallel lines. We show that this number is at least \(\frac{17}{38}n -O(1) \approx 0.4473n\). The best previous bound, \((\sqrt{2}-1)n-O(1)\approx 0.4142n\), which dates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar’s theorem [23] on the number of distinct directions determined by n noncollinear points in the plane.

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References

  1. 1.
    Altman, E.: On a problem of Erdős. American Mathematical Monthly 70, 148–157 (1963)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Altman, E.: Some theorems on convex polygons. Canadian Mathematical Bulletin 15, 329–340 (1972)MathSciNetMATHGoogle Scholar
  3. 3.
    Braß, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)MATHGoogle Scholar
  4. 4.
    Braß, P., Rote, G., Swanepoel, K.J.: Triangles of extremal area or perimeter in a finite planar point set. Discrete & Computational Geometry 26, 51–58 (2001)MathSciNetMATHGoogle Scholar
  5. 5.
    Burton, G.R., Purdy, G.: The directions determined by n points in the plane. Journal of London Mathematical Society 20, 109–114 (1979)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Springer, New York (1991)MATHGoogle Scholar
  7. 7.
    Dumitrescu, A.: On distinct distances from a vertex of a convex polygon. Discrete & Computational Geometry 36, 503–509 (2006)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Dumitrescu, A., Tóth, C.D.: On the number of tetrahedra with minimum, uniform, and distinct volumes in three-space. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 1114–1123. ACM Press, New York (2007)Google Scholar
  9. 9.
    Erdős, P.: On sets of distances of n points. American Mathematical Monthly 53, 248–250 (1946)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Erdős, P., Purdy, G.: Some extremal problems in geometry IV. Congressus Numerantium (Proceedings of the 7th South-Eastern Conference on Combinatorics, Graph Theory, and Computing) 17, 307–322 (1976)Google Scholar
  11. 11.
    Erdős, P., Purdy, G.: Some extremal problems in geometry V. In: Proceedings of the 8th South-Eastern Conference on Combinatorics, Graph Theory, and Computing, pp. 569–578 (1977)Google Scholar
  12. 12.
    Erdős, P., Purdy, G.: Extremal problems in combinatorial geometry. In: Handbook of Combinatorics, vol. 1, pp. 809–874. Elsevier, Amsterdam (1995)Google Scholar
  13. 13.
    Erdős, P., Purdy, G., Straus, E.G.: On a problem in combinatorial geometry. Discrete Mathematics 40, 45–52 (1982)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Goodman, J.E., Pollack, R.: On the combinatorial classification of nondegenerate configurations in the plane. Journal of Combinatorial Theory Ser. A 29, 220–235 (1980)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Goodman, J.E., Pollack, R.: A combinatorial perspective on some problems in geometry. Congressus Numerantium 32, 383–394 (1981)MathSciNetGoogle Scholar
  16. 16.
    Katz, N.H., Tardos, G.: A new entropy inequality for the Erdős distance problem. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 119–126. AMS, Providence (2004)Google Scholar
  17. 17.
    Pach, J.: Personal communication (January 2007)Google Scholar
  18. 18.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. John Wiley, New York (1995)MATHGoogle Scholar
  19. 19.
    Pach, J., Tardos, G.: Isosceles triangles determined by a planar point set. Graphs and Combinatorics 18, 769–779 (2002)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Solymosi, J., Tóth, C.D.: Distinct distances in the plane. Discrete & Computational Geometry 25, 629–634 (2001)MathSciNetMATHGoogle Scholar
  21. 21.
    Straus, E.G.: Some extremal problems in combinatorial geometry. In: Proceedings of the Conference on Combinatorial Theory. Lecture Notes in Mathematics, vol. 686, pp. 308–312. Springer, Heidelberg (1978)Google Scholar
  22. 22.
    Székely, L.: Crossing numbers and hard Erdős problems in discrete geometry. Combinatorics, Probability and Computing 6, 353–358 (1997)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Ungar, P.: 2N noncollinear points determine at least 2N directions. Journal of Combinatorial Theory Ser. A 33, 343–347 (1982)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Csaba D. Tóth
    • 2
  1. 1.Deptartment of Computer Science, University of Wisconsin-Milwaukee, WI 53201-0784USA
  2. 2.Department of Mathematics, MIT, Cambridge, MA 02139USA

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