Discrete & Computational Geometry

, Volume 7, Issue 2, pp 135–152 | Cite as

Convex independent sets and 7-holes in restricted planar point sets

  • Pavel Valtr


For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letcα(k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying\(q(A)< \alpha \sqrt k \) for fixed\(\alpha \geqslant \sqrt {2\sqrt 3 /\pi } \doteq 1.05\), contains at leastc convex independent points. We determine the exact asymptotic behavior ofcα(k), proving that there are two positive constantsβ=β(α),γ such thatβk1/3cα(k)≤γk1/3. To establish the upper bound ofcα(k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying\(q(A)< \alpha \sqrt k \). The construction uses “Horton sets,” which generalize sets without 7-holes constructed by Horton and which have some interesting properties.


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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Pavel Valtr
    • 1
    • 2
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czechoslovakia
  2. 2.Institut für InformatikFreie Universität BerlinBerlin 33Federal Republic of Germany

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