**Hyperspaces**

Theorems 16.1.2 and 16.1.6 were originally proved by Nadler, Quinn, and Stavrakas in [885]. 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.

The topology of the hyperspace \(\mathcal K^n_0\) (Theorem 16.1.7) was discovered by Antonyan and Jonard-Pérez in [32].

On the other hand, hyperspaces of convex bodies of constant width were first studied by Bazilevich and Zarichnyi. They proved in [92] that \(\mathcal W^n\) and other families of convex bodies of constant width are *Q*-manifolds (cf. [87]). 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 [33]).

A related investigation in the infinite-dimensional setting is presented in [93]. The 2-dimensional spherical situation, analogously implying that the mentioned hyperspace is a manifold modeled on the Hilbert cube, is discussed in [89], and in [88] 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). The set of finite sequences in the Hilbert cube is obtained in the particular case of *P* being a square. 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 [91]. And results around the Lie group acting on the hyperspace of constant width bodies inscribed into the unit square are derived in [90].

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). Nevertheless, we can still find some open problems concerning the topology of certain spaces made of equivalence classes of convex sets. This holds, e.g., for the natural equivalence relation induced by the Banach–Mazur distance or by the Gromov–Hausdorff distance. The reader can find more information about these distances and some open problems related to the hyperspaces of convex bodies in [195], [924], and [1131].

**Transnormality**

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. Thus, the reader is also referred to Chapter 11.

In convexity, constant width is usually defined by the property that parallel supporting hyperplanes have constant distance. 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. This viewpoint can be conveniently extended to arbitrary codimensions. More precisely, a closed, connected, smooth submanifold \(M^m\) of \({\mathbb E}^n\) is said to be *transnormal* in \({\mathbb E}^n\) (see [974]) if the affine subspace *A*, which is normal at an arbitrary point of \(M^m\) is also normal at all points where *A* meets \(M^m\). Thus, transnormality generalizes constant width, and it is easy to show that the map *f* from the transnormal manifold to the space of normal (\(n-m\))-flats of it is a covering map. One can continue with *k*-transnormality if *f* is a *k*-fold. For example, the case \(m = 1, n = 3,\) and \(k = 2\) yields a smooth curve *C* embedded in \({\mathbb E}^3\) such that for any \(x \in C\) the normal plane *P*(*x*) meets *C* in a second single point *y* with \(P(x) = P(y)\), and any such curve is a spatial curve of constant width (the converse is generally not true)! Among other results, it was proved in [974] that every compact 2-transnormal manifold is homeomorphic to a sphere (directly generalizing and also including the notion of constant width). In [975], compact *k*-transnormal manifolds are investigated by considering critical points of the distance function from a fixed point of \({\mathbb E}^n\). Many topological results are derived, and it is shown how transnormality is related to the concept of “exact filling” investigated in [977]. Bolton [145] studied connected smooth submanifolds *M* of connected, complete, smooth Riemannian manifolds *R* which are transnormal there. He considered the case where the topology inherited by *M* from *R* is coarser than the manifold topology of *M*, showing that then *M* is dense in *R* and *M* is a leaf of a foliation of *R*. Similarly, Nishikawa [894] determined topological structures of transnormal hypersurfaces in a fairly general Riemannian setting, with different orders of transnormality. Also more from the viewpoint of differential geometry, these structures were studied by him, see [895]. For example, results on the differential geometry of 2-transnormal hypersurfaces in spherical and hyperbolic spaces were obtained, too.

Going back to Euclidean geometry, Irwin [571] proved that there are smooth, simple, closed, and *k*-transnormal curves in \({\mathbb E}^n, n > 3\) with \(k > 2\), and that this is not possible in \({\mathbb E}^3\). Wegner contributed to the notion of transnormality with an impressive series of papers whose sequence we want to discuss now. In [1164], he proved that the projection of a transnormal manifold on its space of normal planes is a covering map. He also verified the following direct generalization of a property of closed convex hypersurfaces of constant width: If two points *x*, *y* of a transnormal manifold have the same normal plane, then (for a suitable choice) the sum of the corresponding radii of principal curvature in the direction of the common double normal equals the distance of *x* and *y*. In [1165], topological results from [975] are extended, and in [1166] it was proved that every transnormal circle is isotopic to a round circle through embeddings that are themselves transnormal, and that every transnormal embedding of the real line must be 1-transnormal. Using related methods, Wegner showed in [1167] that any space curve of constant width 1 can be isotopically deformed through curves of constant width 1 to a curve on a sphere of radius 1, where length, integral curvature, and integral torsion are preserved. Hereby the geodesic curvature is minimized. Nice results from the continuation [1169] are the following ones: a spherical curve of constant width 1 lies between two parallel planes of distance \(\frac{1}{\sqrt{3}}\), and a curve of constant width is isotopic to the circle, where each curve of this deformation process can be chosen as a constant width curve. With the concept of transnormal curves in 3-space, Wegner generalized in [1170] results of Hoschek from [554] and [555] (themselves based on [1167]) on generalized Zindler curves , going also to higher dimensions and using transnormal curves as starting point. Coming back to topological questions from [975], Wegner investigated in [1172] the pencil of the normal spaces of the studied manifold, generalizing also results from [1166]. After the appearance of [1172], Robertson [976] published a nice overview and update on the developments around the transnormality notion, starting with elementary properties of constant width curves, and going on with concepts of transnormality and generating frames, topological questions (transnormal isotopies, etc.), transnormal submanifolds in arbitrary Riemann manifolds, and the relations of transnormal systems to Cartan’s isoparametric hypersurfaces. Wegner continued in [1175] by generalizing results from [213]. He showed that transnormality is not preserved under normal projection onto a hyperplane without the assumption that the manifold has the topological type of a sphere. This implies a new generalized version of the so-called transnormal graph theorem (see [212] and [213]). In [1177] and [1178], the infinitesimal rectangle property (IRP) and rectangle property (RP) of subsets *U* of \({\mathbb E}^n\) are discussed. Namely, *U* has the property (IRP) if, for some \(0 < \varepsilon \le 1\), each rectangle with sides of ratio at most \(\varepsilon \) and having three vertices in *U* must also have its fourth vertex in *U*; and *U* has the property (RP) if this is true for \(\varepsilon = 1\). Wegner extended the known planar situation with (IRP) and (RP) to curves and hypersurfaces in higher dimensions, e.g., proving that closed and simple \(C^2\) curves satisfying (IRP) must be transnormal, and that (RP) yields centrally symmetric spherical curves which are transnormal. On the other hand, any centrally symmetric, spherical curve which is \(C^1\) transnormal satisfies the property (IRP). For a closed smooth hypersurface *U* of positive Gaussian curvature in \({\mathbb E}^n, n > 2,\) and a given affine subspace *A* of dimension \(k, 0< k < n-1\), the set of points in *U*, where the tangent plane of *U* is parallel to *A*, is a differentiable (\(n-k-1\))-submanifold, called the *A*-horizon of *S*. In [1176], it is proved that round spheres are characterized by the property that all horizons of *U* of the same dimension are transnormal. And for \(n=3\) the round sphere is characterized by the existence of two transnormal horizons through every point of *U* if *U* is of constant width. In [1179], Wegner complements results from [216] on self-parallel curves, getting also a new possibility to construct transnormal curves. So the author succeeds in presenting examples of transnormal curves in 4-space with arbitrarily high degree of transnormality, disproving a long-standing conjecture of Irwin [571]. Also the existence of injective infinitely transnormal imbeddings of the real line into the 4-space is verified. Extending nice examples of transnormal submanifolds provided by Wegner in [1179], the authors of [9] present tubular constructions around transnormal submanifolds leading again to transnormal submanifolds.

Two immersions \(f, g: M \rightarrow {\mathbb E}^n\) are called parallel if for every \(x \in M\) the (affine) normal space of *f* at *x* and that of *g* at *x* coincide. The parallel rank of *f* is given by the dimension of the set in the normal space at \(x \in M\), consisting of those points where immersions parallel to *f* pass through. In [335], some properties of these notions, which are also related to the concept of transnormality, are presented. The authors mainly study self-parallel immersions which share several properties with transnormal submanifolds.

Dekster and Wegner [291] successfully carried over the concepts of bodies of constant width (whose boundaries are 2-transnormal topological (\(n-1\))-spheres) and transnormality to spherical *n*-space.

**Homology, Cohomology, and Fiber Bundles**

There are many good books about algebraic topology; we recommend here some of them. The books: “Fiber Bundles” by Steenrod [1088] and “Algebraic Topology” by Spanier [1085] are excellent reference books. Introductory textbooks are: “Introduction to Fiber Bundles” by Porter [948] and “Homology Theory” by Vick [1146]. We also recommend the book [548] by Hocking and Young as an introduction to the theory of homology and cohomology. An excellent reference for the topology of Grassmannians is the paper [946] and of course the classical book [834] by Milnor and Stasheff. An excellent introduction to topology from the viewpoint of discrete mathematics is [734] by Lovász. Finally, two good textbooks referring to differential and algebraic topology are Singer and Thorpe [1069] and Milnor [833].