We'd like to understand how you use our websites in order to improve them. Register your interest.

Cages of Small Length Holding Convex Bodies


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. 1.

    Aberth, O.: An isoperimetric inequality for polyhedra and its application to an extremal problem. Proc. Lond. Math. Soc. 13, 322–336 (1963)

  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)

  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)

  4. 4.

    Bracho, J., Montejano, L., Urrutia, J.: Immobilization of smooth convex figures. Geom. Dedicata 53(2), 119–131 (1994)

  5. 5.

    Coxeter, H.S.M.: Review 1950. Math. Rev. 20, 322 (1959)

  6. 6.

    Czyzowicz, J., Stojmenovic, I., Urrutia, J.: Immobilizing a shape. Int. J. Comput. Geom. Appl. 9(2), 181–206 (1999)

  7. 7.

    Fruchard, A.: Fixing and almost fixing a convex figure (2017). hal-01573119

  8. 8.

    Kovalyov, M.D.: Covering a convex figure by its images under dilatation. Ukrainskij Geom. Sbornik 27/84, 57–68 (1984). (in Russian)

  9. 9.

    Kós, G., Törőcsik, J.: Convex disks can cover their shadow. Discrete Comput. Geom. 5(6), 529–531 (1990)

  10. 10.

    Lillington, J.N.: A conjecture for polytopes. Proc. Cambr. Philos. Soc. 76, 407–411 (1974)

  11. 11.

    Linhart, J.: Kantenlängensumme, mittlere Breite und Umkugelradius konvexer Polytope. Arch. Math. 29, 558–560 (1977)

  12. 12.

    Zamfirescu, T.: Inscribed and circumscribed circles to convex curves. Proc. Am. Math. Soc. 80(3), 455–457 (1980)

Download references


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).

Author information



Corresponding author

Correspondence to Augustin Fruchard.

Additional information

Dedicated to the memory of Ricky Pollack.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Editor in Charge: János Pach

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fruchard, A., Zamfirescu, T. Cages of Small Length Holding Convex Bodies. Discrete Comput Geom (2019).

Download citation


  • Immobilisation
  • Skeleton
  • Steiner tree
  • Convex body

Mathematics Subject Classification

  • 52A15
  • 52A40
  • 52B10