Advertisement

Aequationes mathematicae

, Volume 87, Issue 1–2, pp 43–52 | Cite as

Suitable families of boxes and kernels of staircase starshaped sets in \({\mathbb{R}^d}\)

  • Marilyn Breen
Article

Abstract

Let S be an orthogonal polytope in \({\mathbb{R}^d}\) . There exists a suitable family \({\mathcal{C}}\) of boxes with \({S = \cup \{C : C {\rm in} \mathcal{C}\}}\) such that the following properties hold:
  • The staircase kernel Ker S is a union of boxes in \({\mathcal{C}}\).

    Let \({\mathcal{V}}\) be the family of vertices of boxes in \({\mathcal{C}}\) , and let \({v_o\, \epsilon \mathcal{V}}\) . Point v o belongs to Ker S if and only if v o sees via staircase paths in S every point w in \({\mathcal{V}}\) . Moreover, these staircase paths may be selected to consist of edges of boxes in \({\mathcal{C}}\).

    Let B be a box in \({\mathcal{C}}\) with vertices of B in Ker S. Box B lies in Ker S if and only if, for some b in rel int B and for every translate H of a coordinate hyperplane at \({b, b \epsilon}\) Ker (HS).

    For point p in S, p belongs to Ker S if and only if, for every x in S, there exist some px geodesic λ (p, x) and some corresponding \({\mathcal{C}}\) - chain D containing λ (p, x) such that D is staircase starshaped at p.

Mathematics Subject Classification

Primary 52.A30 52.A35 

Keywords

Orthogonal polytopes Staircase paths Staircase starshaped sets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Breen M.: A Krasnosel’skii-type result for planar sets starshaped via orthogonally convex paths. Periodica Mathematica Hungarica 55, 169–176 (2007)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Breen M.: Characterizing certain staircase convex sets in \({\mathbb{R}^d}\) . Beiträge zur Algebra und Geometrie 51, 251–261 (2010)MATHMathSciNetGoogle Scholar
  3. 3.
    Breen M.: Staircase kernels in orthogonal polygons. Archiv der Mathematik 59, 588–594 (1992)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Breen M.: Using chains of boxes to recognize staircase starshaped sets in \({\mathbb{R}^d}\) . J. Comb. Math. Comb. Comput. (to appear)Google Scholar
  5. 5.
    Chepoi V.: On staircase starshapedness in rectilinear spaces. Geometriae Dedicata 63, 321–329 (1996)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Convexity, Proc. Sympos. Pure Math., vol. 7, pp. 101–180. Amer. Math. Soc., Providence (1962)Google Scholar
  7. 7.
    Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 389–448. North Holland, New York (1993)Google Scholar
  8. 8.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)Google Scholar
  9. 9.
    Lay S.R.: Convex Sets and Their Applications. Wiley, New York (1982)MATHGoogle Scholar
  10. 10.
    Valentine F.A.: Convex Sets. McGraw-Hill, New York (1964)MATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.The University of OklahomaNormanUSA

Personalised recommendations