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Cages of Small Length Holding Convex Bodies

  • Augustin FruchardEmail author
  • Tudor Zamfirescu
Ricky Pollack Memorial Issue
  • 16 Downloads

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.

Keywords

Immobilisation Skeleton Steiner tree Convex body 

Mathematics Subject Classification

52A15 52A40 52B10 

Notes

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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Copyright information

© 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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