Cages of Small Length Holding Convex Bodies

  • Augustin FruchardEmail author
  • Tudor Zamfirescu
Ricky Pollack Memorial Issue


A cage G, defined as the 1-skeleton of a convex polytope in 3-space, holds a compact set K if G cannot move away without meeting the relative interior of K. The main results of this paper establish the infimum of the lengths of cages holding various compact convex sets. First, planar graphs and Steiner trees are investigated. Then the notion of points almost fixing a convex body in the plane is introduced and studied. The last two sections treat cages holding 2-dimensional compact convex sets, respectively the regular tetrahedron.


Immobilisation Skeleton Steiner tree Convex body 

Mathematics Subject Classification

52A15 52A40 52B10 



The authors warmly thank the referee for his/her careful reading. The authors gratefully acknowledge partial support from GDRI ECO-Math. The second author also thanks for the financial support by the NSF of China (11871192).


  1. 1.
    Aberth, O.: An isoperimetric inequality for polyhedra and its application to an extremal problem. Proc. Lond. Math. Soc. 13, 322–336 (1963)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Besicovitch, A.S.: A cage to hold a unit-sphere. In: Proceedings of Symposia in Pure Mathematics, vol. VII, pp. 19–20. American Mathematical Society, Providence (1963)Google Scholar
  3. 3.
    Bracho, J., Fetter, H., Mayer, D., Montejano, L.: Immobilization of solids and mondriga quadratic forms. J. Lond. Math. Soc. 51(1), 189–200 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bracho, J., Montejano, L., Urrutia, J.: Immobilization of smooth convex figures. Geom. Dedicata 53(2), 119–131 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Coxeter, H.S.M.: Review 1950. Math. Rev. 20, 322 (1959)Google Scholar
  6. 6.
    Czyzowicz, J., Stojmenovic, I., Urrutia, J.: Immobilizing a shape. Int. J. Comput. Geom. Appl. 9(2), 181–206 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fruchard, A.: Fixing and almost fixing a convex figure (2017). hal-01573119Google Scholar
  8. 8.
    Kovalyov, M.D.: Covering a convex figure by its images under dilatation. Ukrainskij Geom. Sbornik 27/84, 57–68 (1984). (in Russian)Google Scholar
  9. 9.
    Kós, G., Törőcsik, J.: Convex disks can cover their shadow. Discrete Comput. Geom. 5(6), 529–531 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lillington, J.N.: A conjecture for polytopes. Proc. Cambr. Philos. Soc. 76, 407–411 (1974)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Linhart, J.: Kantenlängensumme, mittlere Breite und Umkugelradius konvexer Polytope. Arch. Math. 29, 558–560 (1977)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zamfirescu, T.: Inscribed and circumscribed circles to convex curves. Proc. Am. Math. Soc. 80(3), 455–457 (1980)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de MathématiquesIRIMAS, Université de Haute-AlsaceMulhouseFrance
  2. 2.Fakultät für MathematikTechnische Universität DortmundDortmundGermany
  3. 3.Institute of Mathematics “Simion Stoilow”Romanian AcademyBucharestRomania
  4. 4.College of MathematicsHebei Normal UniversityShijiazhuangP. R. China

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