Bodies of Constant Width in Topology

Horst Martini (), Luis Montejano and Déborah Oliveros
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Horst Martini: Chemnitz University of Technology, Faculty of Mathematics
Luis Montejano: Universidad Nacional Autónoma de México, Campus Juriquilla, Instituto de Matemáticas
Déborah Oliveros: Universidad Nacional Autónoma de México, Campus Juriquilla, Instituto de Matemáticas

Chapter Chapter 16 in Bodies of Constant Width, 2019, pp 369-398 from Springer

Abstract: Abstract Four sections integrate this chapter. In the first section we shall study the hyperspace of all convex sets in Euclidean space $$\mathbb {E}^n$$ 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 $$\mathbb {E}^3$$ 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.

Date: 2019
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DOI: 10.1007/978-3-030-03868-7_16

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