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  • 1 I: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences II: Department of Mathematics, University College London I: H-1364 Budapest, P.O.B. 127, Hungary II: Gower Street, London Wc1e 6bt, England
  • 2 Institute for Theoretical Computer Science and Department of Applied Mathematics, Charles University Malostranské Nám. 25, 118 00 Praha 1, Czech Republic
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A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

  • Bárány, I. and Füredi, Z., Empty simplices in Euclidean space, Canadian Math. Bull. 30 (1987), 436-445. MR 89g:52004

    'Empty simplices in Euclidean space ' () 30 Canadian Math. Bull. : 436 -445.

  • Bárány, I. and Károlyi, Gy., problems and results around the erdös-szekeres convex polygon theorem, lect. notes comput. sci. 2098 (2001), 91-105.

    'Problems and results around the erdös-szekeres convex polygon theorem ' () 2098 Lect. notes comput. sci. : 91 -105.

    • Search Google Scholar
  • dumitrescu, A., Planar sets with few empty convex polygons, Stud. Sci. Math. Hung. 36 (2000), 93-109. MR 2001f:52037

    'Planar sets with few empty convex polygons ' () 36 Stud. Sci. Math. Hung. : 93 -109.

  • Edelman, P. and Reiner, V., Counting the interior of a point configuration, Discrete Comput. Geom. 23 (2000), 1-13. MR 2000i:52028

    'Counting the interior of a point configuration ' () 23 Discrete Comput. Geom. : 1 -13.

  • Erdös, P., On some problems of elementary and combinatorial geometry, Ann. Mat. Pura. Appl. (4) 103 (1975), 99-108. MR 54#113

    'On some problems of elementary and combinatorial geometry ' () (4) 103 Ann. Mat. Pura. Appl. : 99 -108.

    • Search Google Scholar
  • Erdös, P. and Szekeres, Gy., A combinatorial problem in geometry, Compositio Math. 2 (1935), 464-470.

    'A combinatorial problem in geometry ' () 2 Compositio Math. : 464 -470.

  • Hardy, G. H. and Wright, E. M., An introduction to the Theory of Numbers, 5th ed., Clarendon Press (Oxford) 1979. MR 81i:10002

    An introduction to the Theory of Numbers , ().

  • Horton, J. D., Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), 482-484. MR 85f:52007

    'Sets with no empty convex 7-gons, Canadian Math. ' () 26 Bull. : 482 -484.

  • Katchalski, M. and Meir, A., On empty triangles determined by points in the plane, Acta. Math. Hungar. 51 (1988), 323-328. MR 89f:52021

    'On empty triangles determined by points in the plane ' () 51 Acta. Math. Hungar. : 323 -328.

    • Search Google Scholar
  • Valtr, P., Convex independent sets and 7-holes in restricted planar point sets, Discrete Comput. Geom. 7 (1992), 135-152. MR 93e:52037

    'Convex independent sets and 7-holes in restricted planar point sets ' () 7 Discrete Comput. Geom. : 135 -152.

    • Search Google Scholar
  • Ahrens, C., Gordon, G. and Mcmahon, E. W., Convexity and the beta invariant, Discrete Comput Geom. 22 (1999), 411-424. MR 2000j:52007

    'Convexity and the beta invariant ' () 22 Discrete Comput Geom. : 411 -424.

  • Balog, A. and Deshouillers, J-M., On some convex lattice polytopes, in: Number theory in progress, Volume 2: Elementary and analytic number theory (K. Györy et al., eds.), de Gruyter, Berlin 1999, 591-606. MR 2000f:11083

    Number theory in progress, Volume 2: Elementary and analytic number theory , () 591 -606.

    • Search Google Scholar
  • Valtr, P., On the minimum number of empty polygons in planar point sets, Studia Sci. Math. Hung. 30 (1995), 155-163. MR 96e:52019

    'On the minimum number of empty polygons in planar point sets ' () 30 Studia Sci. Math. Hung. : 155 -163.

    • Search Google Scholar