Advertisement

Journal of Optimization Theory and Applications

, Volume 151, Issue 3, pp 541–551 | Cite as

Blaschke-Type Theorem and Separation of Disjoint Closed Geodesic Convex Sets

  • N. N. Hai
  • P. T. AnEmail author
Article

Abstract

In this paper, we deal with analytic and geometrical properties of geodesic convex sets and geodesic paths. We show that Blaschke’s Theorem for convex sets is also true for geodesic convex sets and geodesic paths in a simple polygon. Some geometrical properties of geodesic triangles are presented. Furthermore, separation of geodesic convex sets is shown.

Keywords

Blaschke’s Theorem Convex sets Geodesic paths Geodesic convex sets Geodesic triangles Separation of convex sets 

References

  1. 1.
    Toussaint, G.T.: Computing geodesic properties inside a simple polygon. Rev. Intell. Artif. 3, 9–42 (1989) Google Scholar
  2. 2.
    Ganguli, A., Cortes, J., Bullo, F.: Multirobot rendezvous with visibility sensors in nonconve environments. IEEE Trans. Robot. 25(2), 340–352 (2009) CrossRefGoogle Scholar
  3. 3.
    Aronov, B., Fortune, S., Wilfong, G.: The furthest-site geodesic voronoi diagram. Discrete Comput. Geom. 9, 217–255 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    An, P.T., Giang, D.T., Hai, N.N.: Some computational aspects of geodesic convex sets in a simple polygon. Numer. Funct. Anal. Optim. 31(4), 221–231 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(1), 611–626 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Suri, S.: Computing geodesic furthest neighbors in simple polygons. J. Comput. Syst. Sci. 39, 220–235 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Rapcsák, T.: Smooth Nonlinear Optimization in ℝn. Kluwer Academic, Dordrecht (1997) Google Scholar
  8. 8.
    O’Rourke, J.: Computational Geometry in C, 2nd edn. Cambridge University Press, Cambridge (1998) zbMATHGoogle Scholar
  9. 9.
    Lee, T., Preparata, F.P.: Euclidean shortest paths in the presence of rectilinear barriers. Networks 14, 393–410 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Berlin (2000) zbMATHGoogle Scholar
  11. 11.
    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier, Amsterdam (2000) CrossRefGoogle Scholar
  12. 12.
    Aubin, J.P.: Applied Abstract Analysis. Wiley, New York (1977) zbMATHGoogle Scholar
  13. 13.
    Webster, R.: Convexity. Oxford University Press, London (1994) zbMATHGoogle Scholar
  14. 14.
    An, P.T.: Method of orienting curves for determining the convex hull of a finite set of points in the plane. Optimization 59(2), 175–179 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    An, P.T.: A modification of Graham’s algorithm for determining the convex hull of a finite planar set. Ann. Math. Inform. 34, 3–8 (2007) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Preparata, F.P., Shamos, M.I.: Computational Geometry—An Introduction, 2nd edn. Springer, Berlin (1988) Google Scholar
  17. 17.
    Benson, R.V.: Euclidean Geometry and Convexity. McGraw-Hill, New York (1966) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsInternational University, Vietnam National UniversityHo Chi Minh CityVietnam
  2. 2.CEMAT, Instituto Superior TécnicoLisbonPortugal
  3. 3.Institute of MathematicsHanoiVietnam

Personalised recommendations