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

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Computing and Combinatorics (COCOON 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2108))

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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).

Supported in part by NSF Grant No. CCR-9821038, and a UCSB-COR faculty research grant.

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References

  1. George E. Andrews An asymptotic expression for the number of solutions of a general class of diophantine equations. Transactions of the AMS 99:272–277, May 1961.

    Article  MATH  Google Scholar 

  2. B. Balog and I. Bárány On the convexh ull of integer points in a disc. DIMACS Series, Discrete and Computational Geometry, 6:39–44, 1991.

    Google Scholar 

  3. I. Bárány, R. Howe, and L. Lovasz On integer points in polyhedra: a lower bound. Combinatorica, 12:135–142, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Bárány and David G. Larman The convexh ull of integer points in a large ball. Math. Ann., 312:167–181, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Cook, M. Hartman, R. Kannan, and C. McDiarmid On integer points in polyhedra. Combinatorica, 12:27–37, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Efrat and C. Gotsman Subpixel image registration using circular fiducials. Internat. J. Comput. Geom. Appl., 4(4):403–422, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Har-Peled An output sensitive algorithm for discrete convexh ulls. Computational Geometry, 10:125–138, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  8. G.H. Hardy and E.M. Wright An introduction to the theory of numbers. Oxford University Press, April 1980.

    Google Scholar 

  9. M.D. Katz and D.J. Volper Data structures for retrieval on square grids. SIAM J. Comput., 15(4):919–931, November 1986.

    Article  MATH  MathSciNet  Google Scholar 

  10. A.M. Vershik On the number of convexlattice polytopes.Geometry and Functional Analysis, 2:381–393, 1992.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara, CA, 93106, USA

    Abdullah N. Arslan & Ömer Eğecioğlu

Authors
  1. Abdullah N. Arslan
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  2. Ömer Eğecioğlu
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Editors and Affiliations

  1. Department of Computer Science, University of Massachusetts, One University Avenue, Lowell, MA, 01854, USA

    Jie Wang

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© 2001 Springer-Verlag Berlin Heidelberg

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Arslan, A.N., Eğecioğlu, Ö. (2001). An Improved Upper Bound on the Size of Planar Convex-Hulls. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_13

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  • DOI: https://doi.org/10.1007/3-540-44679-6_13

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  • Print ISBN: 978-3-540-42494-9

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