A Note on the Number of Empty Triangles

  • Alfredo García
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Let P be a set of n points on the plane, in general position, H of them placed on the boundary of the convex hull of P. In this note we prove that there is a well defined family of empty triangles, the family of empty triangles not generated by an empty convex pentagon, containing exactly n2 − 5n + H + 4 empty triangles. This result immediately implies a slight improvement on the lower bound on the number of empty triangles that every set of n points in the plane must determine.


Empty triangle empty convex pentagon convex hull points in general position 


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  1. 1.
    Aichholzer, O.: [Empty] [colored] k-gons - Recent results on some Erdös-Szekeres type problems. In: Proc. XIII Encuentros de Geometría Computacional, pp. 43–52. Prensas universitarias de Zaragoza, Spain (2009)Google Scholar
  2. 2.
    Bàràny, I., Valtr, P.: Planar point sets with a small number of empty convex polygons. Stud., Sci. Math. Hung. 41(2), 243–269 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bàràny, I., Füredi, Z.: Empty simplices in Euclidean space. Canadian Math. Bull. 30, 436–445 (1987)Google Scholar
  4. 4.
    Dehnhardt, K.: Leere konvexe Vielecke in ebenen Punktmengen, Dissertation, TU Braunschweig (1987)Google Scholar
  5. 5.
    Harborth, H.: Konvexe Fünfecke in ebenen Punktmengen. Elem. Math. 33, 116–118 (1978)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Morris, W., Soltan, V.: The Erdös-Szekeres problem on points in convex position - a survey. Bulletin (new series) of the American Mathematical Society 37(4), 437–458 (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Pinchasi, R., Radoicic, R., Sharir, M.: On empty convex polygons in a planar point set. J. Comb. Theory, Ser. A 113(3), 385–419 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alfredo García
    • 1
  1. 1.Departamento de Métodos Estadísticos and IUMAUniversidad de ZaragozaZaragozaSpain

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