Abstract
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.
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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. 100-0679. 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).
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Alon, N., Katchalski, M. & Pulleyblank, W.R. The maximum size of a convex polygon in a restricted set of points in the plane. Discrete Comput Geom 4, 245–251 (1989). https://doi.org/10.1007/BF02187725
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DOI: https://doi.org/10.1007/BF02187725