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An Improved Upper Bound on the Size of Planar Convex-Hulls

  • Abdullah N. Arslan
  • Ömer Eğecioğlu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2108)

Abstract

Let C be the convex-hull of a set of points S with integral coordinates in the plane. It is well-known that |C| ≤ cD 2/3 for some constant c where D is the diameter of S: i.e. the maximum distance between any pair of points in S. It has been shown that c = 7.559.. for an arbitrary S, and c = 3.496.. in the special case when S is a ball centered at the origin in the plane. In this paper we show that c = 12/ 3v 4p2 = 3.524.. is sufficient for an arbitrary set of lattice points S of diameter D in the plane, and |C| ~ 12 3v2/(9p2) D 2/3 = 3.388..D 2/3 is achieved asymptotically. Our proof is based on the construction of a special set in ?rst quadrant, and the analysis of the result involves the calculation of the average order of certain number-theoretical functions associated with the Euler totient function φ(n).

Keywords

Convex Hull Lattice Point Integer Point Average Order Convex Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Abdullah N. Arslan
    • 1
  • Ömer Eğecioğlu
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA

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