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Bodies of Constant Width in Topology

  • Horst MartiniEmail author
  • Luis Montejano
  • Déborah Oliveros
Chapter

Abstract

Four sections integrate this chapter. In the first section we shall study the hyperspace of all convex sets in Euclidean space \(\mathbb {E}^n\) and, within this, the hyperspace of all bodies of constant width. In the second section, differential and algebraic topology are needed to study an amazing generalization of bodies of constant width known as transnormal manifolds. Section 16.3 is devoted to polyhedra that circumscribe the sphere of diameter 1; within this family, we will characterize those polyhedra which are universal covers. In particular, we will use the theory of fiber bundles to prove that the rhombic dodecahedron circumscribing the sphere of diameter 1 is a universal cover in \(\mathbb {E}^3\). Finally, in Section 16.4 the topology and the geometry of Grassmannian spaces are used to see how big or complicated a collection of constant width sections should be such that the original body is of constant width.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Horst Martini
    • 1
    Email author
  • Luis Montejano
    • 2
  • Déborah Oliveros
    • 2
  1. 1.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Campus JuriquillaQuerétaroMéxico

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