Discrete & Computational Geometry

, Volume 12, Issue 4, pp 439–449 | Cite as

Generalized breadths, circular Cantor sets, and the least area UCC

  • Gy. Elekes


We develop a technique suitable for determining the minimal area convex set that can cover certain prescribed regular polygons. As a side effect we improve the well-known “circle-and-triangle” lower bound on the least area Universal Convex Cover (UCC).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BC]
    K. Bezdek and R. Connelly, Covering curves by translates of a convex set,Amer. Math. Monthly 96 (1989), 789–806.MATHMathSciNetCrossRefGoogle Scholar
  2. [BF]
    T. Bonnensen and W. Fenchel,Theorie der konvexen Körper, Springer-Verlag, Berlin, 1974.CrossRefGoogle Scholar
  3. [D]
    G. F. D. Duff, A smaller universal cover for sets of unit diameter,C. R. Math. Rep. Acad. Sci. Canada 2(1) (1980), 37–42.MATHMathSciNetGoogle Scholar
  4. [H1]
    H. C. Hansen, a small universal cover of figures of unit diameter,Geom. Dedicata 4 (1975), 165–172.MATHCrossRefGoogle Scholar
  5. [H2]
    H. C. Hansen, Small universal covers of sets of unit diameter,Geom. Dedicata 42 (1992), 205–213.MATHMathSciNetCrossRefGoogle Scholar
  6. [J1]
    H. W. E. Jung Über die kleinste Kugel, die eine raumliche Figur einschliesst,Crelles J. reine angew. Math. 123 (1901), 241–257.MATHGoogle Scholar
  7. [J2]
    H. W. E. Jung, Über den kleinsten Kreis, der eine ebene Figur einschliesst,Crelles J. reine angew. Math. 137 (1910), 310–313.MATHGoogle Scholar
  8. [K]
    M. D. Kovalev, A minimal convex covering for triangles,Ukrain. Geom. Sb. 26 (1983), 63–68 (in Russian).MATHMathSciNetGoogle Scholar
  9. [M]
    H. Meschkowski,Ungelöste und unlösbare Probleme der Geometrie, Vieweg, Braunschweig, 1960.MATHCrossRefGoogle Scholar
  10. [N]
    J. C. C. Nitche, The smallest sphere containing a rectifiable curve,Amer. Math. Monthly 78 (1971), 881–882.MathSciNetCrossRefGoogle Scholar
  11. [P1]
    J. Pál, Über ein elementares Variationsproblem,Danske Vidensk. Selsk. Math.-fys. Medd. III 2 (1920), 3–35.Google Scholar
  12. [P2]
    J. Pál, Ein Minimalproblem für Ovale,Math. Ann. 83 (1921), 311–319.MATHMathSciNetCrossRefGoogle Scholar
  13. [S]
    R. Sprague, Über ein elementares Variationsproblem,Mat. Tid. (1936), 96–99.Google Scholar
  14. [T]
    G. Tóth, Private communication.Google Scholar
  15. [W1]
    J. E. Wetzel, Triangular covers for closed curves of constant length,Elem. Math. 25(4) (1970), 78–81.MATHMathSciNetGoogle Scholar
  16. [W2]
    J. E. Wetzel, On Moser’s problem of accommodating closed curves in triangles,Elem. Math. 27(2) (1972), 35–36.MATHMathSciNetGoogle Scholar
  17. [W3]
    J. E. Wetzel, Sectoral covers for curves of constant length,Canad. Math. Bull. 16 (1973), 367–375.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • Gy. Elekes
    • 1
  1. 1.Eötvös University, Múzeum krt. 6-8BudapestHungary

Personalised recommendations