Skip to main content
Springer Nature Link
Log in

Point sets in the unit square and large areas of convex hulls of subsets of points

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

In this paper generalizations of Heilbronn’s triangle problem to convex hulls of j points in the unit square [0,1]2 are considered. By using results on the independence number of linear hypergraphs, for fixed integers k≥3 and any integers n≥k a deterministic o(n 6k−4) time algorithm is given, which finds distributions of n points in [0,1]2 such that, simultaneously for j=3,…,k, the areas of the convex hulls determined by any j of these n points are Ω((log n)1/(j−2)/n (j−1)/(j−2)).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Chazelle B (1989) Lower bounds on the complexity of polytope range searching. J Am Math Soc 2:637–666

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Fundia A (1996) Derandomizing Chebychev’s inequality to find independent sets in uncrowded hypergraphs. Random Struct Algorithms 8:131–147

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Lefmann H (2005) Distributions of points in the unit-square and large k-gons. In: Proceedings 16th ACM-SIAM symposium on discrete algorithms SODA’05. ACM and SIAM, New York, pp 241–250; Long version in Eur J Comb, to appear

    Google Scholar 

  • Lefmann H (2006) Distributions of points and large convex hulls of k points. In: Proceedings second international conference ‘Algorithmic aspects in information and management AAIM’06’. Lectures notes in computer sciences, vol 4041. Springer, Berlin, pp 173–184

    Google Scholar 

  • Lefmann H (2007) Convex hulls of point sets and non-uniform uncrowded hypergraphs. In: Proceedings third international conference ‘Algorithmic aspects in information and management AAIM’07’. Lectures notes in computer sciences, vol 4508. Springer, Berlin, pp 285–295

    Google Scholar 

  • Roth KF (1951) On a problem of Heilbronn. J Lond Math Soc 26:198–204

    Article  MATH  Google Scholar 

  • Roth KF (1972a) On a problem of Heilbronn, II. Proc Lond Math Soc (3) 25:193–212

    Article  MATH  Google Scholar 

  • Roth KF (1972b) On a problem of Heilbronn, III. Proc Lond Math Soc (3) 25:543–549

    Article  MATH  Google Scholar 

  • Roth KF (1973) Estimation of the area of the smallest triangle obtained by selecting three out of n points in a disc of unit area. In: Proceedings of the symposia in pure mathematics, vol 24. AMS, Providence, pp 251–262

    Google Scholar 

  • Roth KF (1976) Developments in Heilbronn’s triangle problem. Adv Math 22:364–385

    Article  MATH  Google Scholar 

  • Shaikhet A (2007) The on-line Heilbronn’s triangle problem in d dimensions. MSc thesis, Department of Computer Science, The Technion, Haifa, Israel

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

    Article  MATH  Google Scholar 

  • Spencer J (1972) Turán’s theorem for k-graphs. Discrete Math 2:183–186

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Informatik, TU Chemnitz, 09107, Chemnitz, Germany

    Hanno Lefmann

Authors
  1. Hanno Lefmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hanno Lefmann.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lefmann, H. Point sets in the unit square and large areas of convex hulls of subsets of points. J Comb Optim 16, 182–195 (2008). https://doi.org/10.1007/s10878-008-9168-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-008-9168-7

Keywords

Search

Navigation