Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Distributions of Points in d Dimensions and Large k-Point Simplices

  • 91 Accesses

  • 2 Citations

Abstract

We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with kd distributions of n points in the d-dimensional unit cube [0,1]d, such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ k,d(n), the supremum of this minimum volume over all distributions of n points in [0,1]d, we show that c k,d⋅(log n)1/(dk+1)/n k/(dk+1)Δ k,d(n)≤c k,d′/n k/d for fixed 2≤kd, and, moreover, for odd integers k≥1, we show the upper bound Δ k,d(n)≤c k,d″/n k/d+(k−1)/(2d(d−1)), where c k,d,c k,d′,c k,d″>0 are constants.

References

  1. 1.

    Ajtai, M., Komlós, J., Pintz, J., Spencer, J., Szemerédi, E.: Extremal uncrowded hypergraphs. J. Comb. Theory Ser. A 32, 321–335 (1982)

  2. 2.

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

  3. 3.

    Barequet, G.: The on-line Heilbronn’s triangle problem. Discrete Math. 283, 7–14 (2004)

  4. 4.

    Barequet, G., Naor, J.: Large kD simplices in the D-dimensional unit cube. Far East J. Appl. Math. 24, 343–354 (2006)

  5. 5.

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

  6. 6.

    Bertram-Kretzberg, C., Lefmann, H.: The algorithmic aspects of uncrowded hypergraphs. SIAM J. Comput. 29, 201–230 (1999)

  7. 7.

    Bertram-Kretzberg, C., Hofmeister, T., Lefmann, H.: An algorithm for Heilbronn’s problem. SIAM J. Comput. 30, 383–390 (2000)

  8. 8.

    Brass, P.: An upper bound for the d-dimensional Heilbronn triangle problem. SIAM J. Discrete Math. 19, 192–195 (2005)

  9. 9.

    Duke, R.A., Lefmann, H., Rödl, V.: On uncrowded hypergraphs. Random Struct. Algorithms 6, 209–212 (1995)

  10. 10.

    Jiang, T., Li, M., Vitany, P.: The average case area of Heilbronn-type triangles. Random Struct. Algorithms 20, 206–219 (2002)

  11. 11.

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

  12. 12.

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

  13. 13.

    Lefmann, H.: On Heilbronn’s problem in higher dimension. Combinatorica 23, 669–680 (2003)

  14. 14.

    Lefmann, H.: Large triangles in the d-dimensional unit-cube. Theor. Comput. Sci. 363, 85–98 (2006)

  15. 15.

    Lefmann, H.: Distributions of points in the unit-square and large k-gons. In: Proc. 16th Symposium on Discrete Algorithms SODA’2005, pp. 241–250. ACM and SIAM, Eur. J. Comb. (to appear)

  16. 16.

    Lefmann, H., Schmitt, N.: A deterministic polynomial time algorithm for Heilbronn’s problem in three dimensions. SIAM J. Comput. 31, 1926–1947 (2002)

  17. 17.

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

  18. 18.

    Roth, K.F.: On a problem of Heilbronn, II and III. Proc. Lond. Math. Soc. 25(3), 193–212 and 543–549 (1972)

  19. 19.

    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. In: Proc. of Symposia in Pure Mathematics, vol. 24, pp. 251–262. Am. Math. Soc., Providence (1973)

  20. 20.

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

  21. 21.

    Shaikhet, A.: The on-line Heilbronn’s triangle problem in d dimensions. M.Sc. Thesis, Department of Computer Science, The Technion, Haifa, Israel (2007)

  22. 22.

    Schmidt, W.M.: On a problem of Heilbronn. J. Lond. Math. Soc. 4(2), 545–550 (1972)

Download references

Author information

Correspondence to Hanno Lefmann.

Additional information

A preliminary version of this paper appeared in COCOON ’05.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lefmann, H. Distributions of Points in d Dimensions and Large k-Point Simplices. Discrete Comput Geom 40, 401–413 (2008). https://doi.org/10.1007/s00454-007-9041-y

Download citation

Keywords

  • Heilbronn’s triangle problem
  • Hypergraphs
  • Independence number