Convexity in Discrete Space

  • Anthony J. Roy
  • John G. Stell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2825)


This paper looks at Coppel’s axioms for convexity, and shows how they can be applied to discrete spaces. Two structures for a discrete geometry are considered: oriented matroids, and cell complexes. Oriented matroids are shown to have a structure which naturally satisfies the axioms for being a convex geometry. Cell complexes are shown to give rise to various different notions of convexity, one of which satisfies the convexity axioms, but the others also provide valid notions of convexity in particular contexts. Finally, algorithms are investigated to validate the sets of a matroid, and to compute the convex hull of a subset of an oriented matroid.


Convexity axioms alignment spaces affine spaces convex spaces convex hull discrete geometry oriented matroids cell complexes matroid algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BLS+99.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.: Oriented Matroids. Encyclopedia of Mathematics and its Applications, vol. 46. CUP (1999)Google Scholar
  2. BY98.
    Boissonnat, J.-D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  3. CB+97.
    Cohn, A.G., Bennett, B., et al.: Qualitative spatial representation and reasoning with the region connection calculus. GeoInformatica 1, 275–316 (1997)CrossRefGoogle Scholar
  4. Cop98.
    Coppel, W.A.: Foundations of Convex Geometry. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  5. dvOS99.
    de Berg, M., van Krevald, M., Overmars, M., Schwarzkopf, O.: Computational Geometry Algorithms and Applications. Springer, Heidelberg (1999)Google Scholar
  6. Eck01.
    Eckhardt, U.: Digital lines and digital convexity. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 209–228. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. Gal99.
    Galton, A.: The mereotopology of discrete space. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 251–266. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. Knu92.
    Knuth, D.E.: Axioms and Hulls. LNCS, vol. 606. Springer, Heidelberg (1992)zbMATHGoogle Scholar
  9. Kov89 .
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics and Image Processing 46, 141–161 (1989)CrossRefGoogle Scholar
  10. Kov92.
    Kovalevsky, V.A.: Finite topology and image analysis. Advances in Electronics and Electron Physics 84, 197–259 (1992)Google Scholar
  11. MV99.
    Masolo, C., Vieu, L.: Atomicity vs Infinite divisibility of space. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 235–250. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. Oxl92.
    Oxley, J.G.: Matroid Theory. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (1992)zbMATHGoogle Scholar
  13. Ros79.
    Rosenfeld, A.: Digital topology. American Mathematical Monthly 86(8), 621–630 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  14. RS02.
    Roy, A.J.O., Stell, J.G.: A qualitative account of discrete space. In: Egenhofer, M.J., Mark, D.M. (eds.) GIScience 2002. LNCS, vol. 2478, pp. 276–290. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. Ste00.
    Stell, J.G.: The representation of discrete multi-resolution spatial knowledge. In: Cohn, A.G., Giunchiglia, F., Selman, B. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of KR 2000, pp. 38–49. Morgan Kaufmann, San Francisco (2000)Google Scholar
  16. von75 .
    von Randow, R.: Introduction to the Theory of Matroids. Lecture Notes in Economics and Mathematical Systems. Springer, Heidelberg (1975)zbMATHGoogle Scholar
  17. Web94.
    Roger Webster. Convexity. OUP (1994)Google Scholar
  18. Web01.
    Webster, J.: Cell complexes and digital convexity. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 272–284. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. Webar.
    Webster, J.: Cell complexes, oriented matroids and digital geometry. Theoretical Computer Science (to appear)Google Scholar
  20. Wel76.
    Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)zbMATHGoogle Scholar
  21. WF00.
    Winter, S., Frank, A.U.: Topology in raster and vector representation. Geoinformatica 4(1), 35–65 (2000)zbMATHCrossRefGoogle Scholar
  22. Win95.
    Winter, S.: Topological relations between discrete regions. LNCS, vol. 951, pp. 310–327 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Anthony J. Roy
    • 1
  • John G. Stell
    • 1
  1. 1.School of ComputingUniversity of LeedsLeedsUK

Personalised recommendations