Skip to main content
SpringerLink
Log in

On the Convex Hull of the Integer Points in a Bi-circular Region

  • Conference paper
Combinatorial Image Analysis (IWCIA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5852))

Included in the following conference series:

  • 735 Accesses

Abstract

Given a set S ⊆ℝ2, denote \(S_{\mathbb {Z}} = S \cap \mathbb {Z}^2\). We obtain bounds for the number of vertices of the convex hull of S ℤ, where S ⊆ℝ2 is a convex region bounded by two circular arcs. Two of the bounds are tight bounds—in terms of arc length and in terms of the width of the region and the radii of the circles, respectively. Moreover, an upper bound is given in terms of a new notion of “set oblongness.” The results complement the well-known O(r 2/3) bound [2] which applies to a disc of radius r.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Acketa, D.M., Žunić, J.: On the Maximal Number of Edges of Convex Digital Polygons Included into a m ×m-Grid. Journal of Combinatorial Theory (Ser. A) 69, 358–368 (1995)

    Article  MATH  Google Scholar 

  2. Balog, A., Bárány, I.: On the Convex Hull of the Integer Points in a Disc. DIMACS Series, vol. 6, pp. 39–44 (1991)

    Google Scholar 

  3. Bárány, I.: Random points and lattice points in convex bodies. Bulletin of the American Mathematical Society 45(3), 339–365 (2008)

    Article  MATH  Google Scholar 

  4. Bárány, I., Hove, R., Lovász, L.: On Integer Points in Polyhedra: A Lower Bound. Combinatorica 12(2), 135–142 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bárány, I., Larman, D.G.: The Convex Hull of the Integer Points in a Large Ball. Math. Ann. 312, 167–181 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Boltianskii, V.G., Soltan, P.S.: Combinatorial Geometry of Various Classes of Convex Sets, Stiinţa, Chişinău (1978)

    Google Scholar 

  7. Brimkov, V.E.: Digitization Scheme that Assures Faithful Reconstruction of Plane Figures. Pattern Recognition 42, 1637–1649 (2009)

    Article  MATH  Google Scholar 

  8. Brimkov, V.E., Barneva, R.P.: On the Polyhedral Complexity of the Integer Points in a Hyperball. Theoretical Computer Science 406, 24–30 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brimkov, V.E., Dantchev, S.S.: Real Data – Integer Solution Problems within the Blum-Shub-Smale Computational Model. Journal of Complexity 13, 279–300 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chirkov, A.Y.: On the Relation of Upper Bounds on the Number of Vertices of the Convex Hull of the Integer Points of a Polyhedron with its Metric Characteristics (in Russian). In: Alekseev, V.E., et al. (eds.) 2nd Int. Conf. Mathematical Algorithms, pp. 169–174. Nizhegorod Univ. Press (1997)

    Google Scholar 

  11. Garey, M.S., Johnson, D.S.: Computers and Intractability: a Guide to the Theory of NP-Completeness. Freeman & Co., San Francisco (1979)

    MATH  Google Scholar 

  12. Hayes, A.S., Larman, D.C.: The Vertices of the Knapsack Polytope. Discrete Applied Mathematics 6, 135–138 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jarnik, V.: Über Gitterpunkte and konvexe Kurven. Math. Zeitschrift 24, 500–518 (1925)

    Article  MATH  MathSciNet  Google Scholar 

  14. McMullen, P.: The Maximum Numbers of Facets of a Convex Polytope. Mathematika 17, 179–184 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  15. Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization. Prentice-Hall, New Jersey (1982)

    MATH  Google Scholar 

  16. Preparata, F., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985)

    Google Scholar 

  17. Rubin, D.S.: On the Unlimited Number of Faces in Integer Hulls of Linear Programs with a Single Constraint. Operations Research 18(5), 940–945 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  18. Schmidt, W.M.: Integer Points on Curves and Surfaces. Monatshefte für Math. 99, 45–72 (1985)

    Article  MATH  Google Scholar 

  19. Shevchenko, V.N.: On the Number of Extremal Points in Integer Programming. Kibernetika 4, 133–134 (1981)

    MathSciNet  Google Scholar 

  20. Valentine, F.A.: Convex Sets. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  21. van der Corupt, J.G.: Verschärfung der Abschätzung beim Teilerproblem. Math. Annalen 87, 39–65 (1922)

    Article  Google Scholar 

  22. van der Vel, M.: Theory of Convex Structures. Elsevier, Amsterdam (1993)

    MATH  Google Scholar 

  23. Veselov, S.I., Chirkov, A.Y.: Bounds on the Number of Vertices of Integer Polyhedra (in Russian). Discrete Analysis and Operations Research 14(2), 14–31 (2007)

    MathSciNet  Google Scholar 

  24. Žunić, J.: Notes on Optimal Convex Lattice Polygons. Bull. London Math. Society 30, 377–385 (1998)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, SUNY Buffalo State College, Buffalo, NY 14222, USA

    Valentin E. Brimkov

Authors
  1. Valentin E. Brimkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Automatic Control, Centro de Investigación y de Estudios Avanzados (CINVESTAV- IPN), Av. I.P.N. 2508, Col. San Pedro Zacatenco, 07000 D.F., Mexico

    Petra Wiederhold

  2. Department of Computer Science, State University of New York at Fredonia, NY 14063, Fredonia,, USA

    Reneta P. Barneva

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Brimkov, V.E. (2009). On the Convex Hull of the Integer Points in a Bi-circular Region. In: Wiederhold, P., Barneva, R.P. (eds) Combinatorial Image Analysis. IWCIA 2009. Lecture Notes in Computer Science, vol 5852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10210-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10210-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10208-0

  • Online ISBN: 978-3-642-10210-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation