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.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Aberth, O.: An isoperimetric inequality for polyhedra and its application to an extremal problem. Proc. Lond. Math. Soc. 13, 322–336 (1963)
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)
Bracho, J., Fetter, H., Mayer, D., Montejano, L.: Immobilization of solids and mondriga quadratic forms. J. Lond. Math. Soc. 51(1), 189–200 (1995)
Bracho, J., Montejano, L., Urrutia, J.: Immobilization of smooth convex figures. Geom. Dedicata 53(2), 119–131 (1994)
Coxeter, H.S.M.: Review 1950. Math. Rev. 20, 322 (1959)
Czyzowicz, J., Stojmenovic, I., Urrutia, J.: Immobilizing a shape. Int. J. Comput. Geom. Appl. 9(2), 181–206 (1999)
Fruchard, A.: Fixing and almost fixing a convex figure (2017). hal-01573119
Kovalyov, M.D.: Covering a convex figure by its images under dilatation. Ukrainskij Geom. Sbornik 27/84, 57–68 (1984). (in Russian)
Kós, G., Törőcsik, J.: Convex disks can cover their shadow. Discrete Comput. Geom. 5(6), 529–531 (1990)
Lillington, J.N.: A conjecture for polytopes. Proc. Cambr. Philos. Soc. 76, 407–411 (1974)
Linhart, J.: Kantenlängensumme, mittlere Breite und Umkugelradius konvexer Polytope. Arch. Math. 29, 558–560 (1977)
Zamfirescu, T.: Inscribed and circumscribed circles to convex curves. Proc. Am. Math. Soc. 80(3), 455–457 (1980)
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).
Dedicated to the memory of Ricky Pollack.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Editor in Charge: János Pach
About this article
Cite this article
Fruchard, A., Zamfirescu, T. Cages of Small Length Holding Convex Bodies. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00144-4
- Steiner tree
- Convex body
Mathematics Subject Classification