More precisely, a closed, connected, smooth submanifold \(M^m\) of \(^n\) are called parallel if for every \(x \in M\) the (affine) normal space of f at x and that of g at x coincide. This viewpoint can be conveniently extended to arbitrary codimensions. For compact, connected and smooth hypersurfaces this is equivalent to the property that a chord being normal at one of its endpoints is also normal at the other one, yielding a double normal. In convexity, constant width is usually defined by the property that parallel supporting hyperplanes have constant distance. Thus, the reader is also referred to Chapter 11. In this subsection we want to confirm that the concept of transnormality, as direct generalization of the constant width notion, is a powerful tool to obtain deep results (mainly) in differential geometry. The reader can find more information about these distances and some open problems related to the hyperspaces of convex bodies in , , and. This holds, e.g., for the natural equivalence relation induced by the Banach–Mazur distance or by the Gromov–Hausdorff distance. Nevertheless, we can still find some open problems concerning the topology of certain spaces made of equivalence classes of convex sets. Nowadays, we can find in the literature numerous results about the topology of different kinds of hyperspaces of closed convex sets (in finite and infinite dimensions). And results around the Lie group acting on the hyperspace of constant width bodies inscribed into the unit square are derived in. The hyperspace of spatial curves of constant width whose plane projection is a circle is homeomorphic to the product of the Hilbert cube and a line, see. The set of finite sequences in the Hilbert cube is obtained in the particular case of P being a square. The 2-dimensional spherical situation, analogously implying that the mentioned hyperspace is a manifold modeled on the Hilbert cube, is discussed in, and in it is proved that the hyperspace of all rotors (respectively, of all smooth rotors, see Section 17.1) in a regular polygon P is homeomorphic to the Hilbert cube (respectively, to the separable Hilbert space). Later, Antonyan, Jonard-Pérez, and Juárez-Ordoñez gave a complete topological description of \(\mathcal W^n\) and \(\mathcal W^n_0\), among other subspaces of \(\mathcal W^n\) (see ).Ī related investigation in the infinite-dimensional setting is presented in. They proved in that \(\mathcal W^n\) and other families of convex bodies of constant width are Q-manifolds (cf. On the other hand, hyperspaces of convex bodies of constant width were first studied by Bazilevich and Zarichnyi. The topology of the hyperspace \(\mathcal K^n_0\) (Theorem 16.1.7) was discovered by Antonyan and Jonard-Pérez in. The proof of Theorem 16.1.2 presented in this book is a slight modification of the original one, whereas the proof of Theorem 16.1.6 is completely different. Further entries in this table can be computed from the information above together with the table of stable homotopy groups of spheres.Theorems 16.1.2 and 16.1.6 were originally proved by Nadler, Quinn, and Stavrakas in. The first exotic spheres were constructed by John Milnor ( 1956) in dimension n = 7. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic"). In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
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