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## Über dieses Buch

This is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these concepts.
An introduction to convexity theory is first provided, and the basic properties of constant width bodies are then presented. The book then delves into a number of related topics, which include Constant width bodies in convexity (sections and projections, complete and reduced sets, mixed volumes, and further partial fields)
Sets of constant width in non-Euclidean geometries (in real Banach spaces, and in hyperbolic, spherical, and further non-Euclidean spaces)
The concept of constant width in analysis (using Fourier series, spherical integration, and other related methods)
Sets of constant width in differential geometry (using systems of lines and discussing notions like curvature, evolutes, etc.)
Bodies of constant width in topology (hyperspaces, transnormal manifolds, fiber bundles, and related topics)
The notion of constant width in discrete geometry (referring to geometric inequalities, packings and coverings, etc.)
Technical applications, such as film projectors, the square-hole drill, and rotary engines

Bodies of Constant Width: An Introduction to Convex Geometry with Applications will be a valuable resource for graduate and advanced undergraduate students studying convex geometry and related fields. Additionally, it will appeal to any mathematicians with a general interest in geometry.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The circle is the geometric locus of all points equidistant from a fixed point called its center. It is precisely due to this property that wheels are round or circular in shape. An axle placed at the center of the wheel does not move up and down when the circle turns. It only moves laterally at a constant height from the ground, and this is because every ray of the circle from the axle to the edge of the wheel has the same length.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 2. Convex Geometry

Abstract
Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 3. Basic Properties of Bodies of Constant Width

Abstract
One of the essential characteristics of bodies of constant width is that, like balls, they have a diameter in every direction. Diameters are those chords of a convex body that have maximum length, and it is their behavior which gives constant width bodies their basic properties. Unlike the diameters of a ball, those of a body of constant width do not always meet at a single point, but when they do so, it is because the body is indeed a ball.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 4. Figures of Constant Width

Abstract
In this chapter, bodies of constant width in the plane are studied. We call them figures of constant width. In studying them, it is important to recall from Section 3.​1 that the concepts “normal”, “binormal”, “diameter”, and “diametral chord” coincide.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 5. Systems of Lines in the Plane

Abstract
A system of lines consists of an assignment of a line in the plane for any direction. In this section, we will be interested in studying systems of lines, in particular those which are combined with a given convex set by a certain property. For example, the system of lines that leave a fixed proportion of area, or a fixed proportion of perimeter, in one side of the convex set for every direction. Consider, for instance, the collection of tangent lines of a given strictly convex set in the plane which varies continuously with respect an angle, or a system of diametral lines, a system of median lines, etc. The purpose of this chapter is to study some useful and important systems of lines.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 6. Spindle Convexity

Abstract
Let h be a positive real number and let p and q be two points in Euclidean n-space $$\mathbb {E}^n$$, no more than 2h apart. The h-interval determined by this pair of points is the intersection of all balls of radius h that contain p and q. We say that a set $$\phi$$, with diameter less than or equal to 2h, is spindle h-convex if given a pair of points p and q in $$\phi$$, the h-interval they determine is also in $$\phi$$.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 7. Complete and Reduced Convex Bodies

Abstract
We say that a compact set in $$\mathbb {E}^n$$ is complete (or diametrically complete) if, adding any point to it, its diameter will increase. If we take the partially ordered set $$\Omega _h$$ of all compact sets of diameter h in n-dimensional Euclidean space ordered by inclusion, complete bodies are precisely the maximal elements of $$\Omega _h$$. That is, a compact set A in $$\Omega _h$$ is a maximal element of $$\Omega _h$$, or a complete body, if A is equal to B whenever A is contained in B, for B in $$\Omega _h$$. The two main results of this chapter are that complete bodies are precisely bodies of constant width h, and that every element of $$\Omega _h$$ is contained in a maximal body; that is, that it can be completed to a body of constant width. These results are known as the Theorems of Meissner and Pál, respectively. Section 7.4 will be devoted to the study of reduced convex bodies, a notion somehow “dual” to completeness, and in Section 7.5 we complete convex bodies preserving some of their original characteristics, such as symmetries.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 8. Examples and Constructions

Abstract
This chapter is dedicated to concrete examples of constant width sets and procedures on how to construct them. The most notorious convex body of constant width is undoubtedly the Reuleaux triangle of width h which is the intersection of three disks of radius h and whose boundary consists of three congruent circular arcs of radius h. In Section 8.1, we will see that the Reuleaux triangle can be generalized to plane convex figures of constant width h whose boundary consists of a finite number of circular arcs of radius h. They are called Reuleaux polygons. The plan for the rest of the chapter is the following: In Section 8.2, we will study the 3-dimensional analogue of the Reuleaux triangle, and in Section 8.3, we will construct Meissner’s mysterious bodies from it. In fact, in this section, we will use the concepts of ball polytope and Reuleaux polytope to construct 3-dimensional bodies of constant width with the help of special embeddings of self-dual graphs. In Section 8.4, we will give a procedure of finitely many steps to construct 3-dimensional constant width bodies from Reuleaux polygons, and in Section 8.5, we will construct constant width bodies with analytic boundaries.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 9. Sections of Bodies of Constant Width

Abstract
In Chapter 3 it was proven that the property of constant width is inherited under orthogonal projection but not under sections. The proof of this fact was not a constructive one, that is, no nonconstant width section of a body of constant width was actually exhibited. In fact, it was proven that if all sections of a convex body have constant width, then the body is a ball. Since there are bodies of constant width other than the ball, it was concluded that they must all have at least one section that is not of constant width. To show this could, however, be tricky, even in cases as simple as the body produced by rotating the Reuleaux triangle around one of its axes of symmetry.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 10. Bodies of Constant Width in Minkowski Spaces

Abstract
In Euclidean space, the length of a segment depends only on its magnitude, never on its direction. However, for certain geometrical problems the need arises to give a different definition for the length of a segment that depends on both the magnitude and the direction.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 11. Bodies of Constant Width in Differential Geometry

Abstract
Let $$\phi \subset \mathbb {E}^n$$ be a strictly convex body whose boundary is twice differentiable and whose curvature never vanishes. Recall that the inverse Gauss map $$\gamma :\mathbb {S}^{n-1} \rightarrow {{\,\mathrm{\mathrm {bd}}\,}}\phi$$ is a diffeomorphism that assigns to each unit vector $$u\in \mathbb {S}^{n-1}$$ the point $$\gamma (u)$$ in the boundary of $$\phi$$ for which u is the outward unit normal vector.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 12. Mixed Volumes

Abstract
The notion of mixed volumes represents a profound concept first discovered by Minkowski in 1900. In the letter [838] he wrote to Hilbert explaining his discoveries as interesting and quite enlightening. As we can see below, this concept will allow us to prove several classical theorems on the volume of constant width bodies in a somewhat unexpected way.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 13. Bodies of Constant Width in Analysis

Abstract
One of the most fascinating theorems on 3-dimensional bodies of constant width, stated and proved by H. Minkowski in 1904, is presented in this section.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 14. Geometric Inequalities

Abstract
Our first isoperimetric inequality is the following (see Theorem 2.​7.​1): For every plane convex body $$\phi \subset \mathbb {E}^2$$ of area A and perimeter P we have
$$P^2-4A\pi \ge 0,$$
and equality holds only for 2-dimensional disks.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 15. Bodies of Constant Width in Discrete Geometry

Abstract
We start with the versions of the Helly’s Theorem developed by V. Klee [628]. Let $$\phi$$ and $$\psi$$ be two convex bodies in $$\mathbb {E}^n$$, and consider the following two subsets:
\begin{aligned} \{x\in \mathbb {E}^n&\mid x + \phi \subset \psi \},\\ \{x\in \mathbb {E}^n&\mid x + \phi \supset \psi \}. \end{aligned}
It is easy to see that both sets are convex bodies. From this, the following variant of Helly’s theorem is immediately obtained.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 16. Bodies of Constant Width in Topology

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.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 17. Concepts Related to Constant Width

Abstract
A polytope P is circumscribed about a convex body $$\phi \subset \mathbb {E}^n$$ if $$\phi \subset P$$ and each facet of P intersects $$\phi$$; i.e., every facet of P is contained in a support hyperplane of $$\phi$$. A polytope P is inscribed in the convex body $$\phi$$ if $$P\subset \phi$$ and each of its vertices belongs to $${{\,\mathrm{\mathrm {bd}}\,}}\phi$$.
Horst Martini, Luis Montejano, Déborah Oliveros

### Chapter 18. Bodies of Constant Width in Art, Design, and Engineering

Abstract
As we mentioned before, a normal wheel (rotating around a fixed axis) must be circular in shape to allow smooth forward motion without vertical bumps. However, a cylindrical roller does not require a circular cross section to allow smooth forward motion. In fact, any curve of constant width, such as the Reuleaux triangle, may be used as a cylindrical roller. Besides this nice property, the curves of constant width, in particular, the Reuleaux triangle, have been exploited by engineers, artists, and designers to obtain wonderful objects and number of ingenious mechanisms.
Horst Martini, Luis Montejano, Déborah Oliveros

### Backmatter

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