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Cages of Small Length Holding Convex Bodies

Abstract

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.

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Acknowledgements

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).

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Correspondence to Augustin Fruchard.

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Dedicated to the memory of Ricky Pollack.

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Fruchard, A., Zamfirescu, T. Cages of Small Length Holding Convex Bodies. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00144-4

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Keywords

  • Immobilisation
  • Skeleton
  • Steiner tree
  • Convex body

Mathematics Subject Classification

  • 52A15
  • 52A40
  • 52B10