On empty convex polygons in a planar point set

Rom Pinchasi*, Radoš Radoičić, Micha Sharir

*Corresponding author for this work

Research output: Contribution to conferencePaperpeer-review

1 Scopus citations


Let P be a set of n points in general position in the plane. Let X k(P) denote the number of empty convex k-gons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities Xk(P) and several related quantities. Most of these equalities and inequalities are new, except for a couple that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, algebraic topology and commutative algebra. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.

Original languageEnglish
Number of pages10
StatePublished - 2004
EventProceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) - Brooklyn, NY, United States
Duration: 9 Jun 200411 Jun 2004


ConferenceProceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04)
Country/TerritoryUnited States
CityBrooklyn, NY


FundersFunder number
Israel Science Fund
US—Israeli Binational Science Foundation
National Science FoundationCCR-00-98246, CCR-97-32101
Tel Aviv University


    • Continuous motion
    • Empty hexagon
    • Empty k-gon


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