Find out how to access previewonly content
The maximum size of a convex polygon in a restricted set of points in the plane
 N. Alon,
 M. Katchalski,
 W. R. Pulleyblank
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Assume we havek points in general position in the plane such that the ratio between the maximum distance of any pair of points to the minimum distance of any pair of points is at mostα√k, for some positive constantα. We show that there exist at leastβk ^{1/4} of these points which are the vertices of a convex polygon, for some positive constantβ=β(α). On the other hand, we show that for every fixedε>0, ifk>k(ε), then there is a set ofk points in the plane for which the above ratio is at most 4√k, which does not contain a convex polygon of more thank ^{1/3+ε } vertices.
The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 1000679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project “Combinatorial Optimization” of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).
 Bateman, P., Erdös, P. (1951) Geometrical extrema suggested by a lemma of Besicovitch. Amer. Math. Monthly 58: pp. 306314
 Erdös, P. (1973) The Art of Counting. MIT Press, Cambridge, MA
 Erdös, P., Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Math. 2: pp. 463470
 Erdös, P., Szekeres, G. (1960) On some extremum problems in elementary geometry. Ann. Univ. Sci. Budapest 3/4: pp. 5362
 Hadwiger, H., Debrunner, H., Klee, V. (1964) Combinatorial Geometry in the Plane. Holt, Rinehart, Winston, New York
 Horton, J. D. (1983) Sets with no empty convex 7gons. Canad. Math. Bull. 26: pp. 482484
 Rademacher, H., Toeplitz, O. (1957) The Enjoyment of Mathematics. Princeton University Press, Princeton, NJ
 Title
 The maximum size of a convex polygon in a restricted set of points in the plane
 Journal

Discrete & Computational Geometry
Volume 4, Issue 1 , pp 245251
 Cover Date
 19891201
 DOI
 10.1007/BF02187725
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 N. Alon ^{(1)}
 M. Katchalski ^{(2)}
 W. R. Pulleyblank ^{(3)}
 Author Affiliations

 1. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel
 2. Department of Mathematics, TechnionIsrael Institute of Technology, Haifa, Israel
 3. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada