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.

In the great garden of Geometry, everyone can pick up a bunch of flowers, simply following his taste.

David Hilbert

Notes

One can see in the text above that curves and bodies of constant width have, as famous geometric figures, many inspiring properties, aspects, and applications. Even more: behind a seemingly simple and “narrow” definition of a class of geometric figures an unexpectedly large variety of properties and applications is “hidden”, not visible at first glance and discussed here in view of more than thousand (directly or indirectly related) mathematical references! This is sufficiently potential to give the non-expert also some feeling how intuitive and refreshing notions and methods from convexity are, and how beautiful this field of geometry is. Thus, constant width sets might play a very important role in the following sense. They present an excellent starting field to become familiar with convex geometry! Even the authors were surprised when they observed how many research papers related to this topic exist! We will try to reflect the widespread occurrence of constant width sets in the literature, partially also in a compressed way, i.e., the vast amount of references on certain aspects of constant width sets is mainly concentrated in concise notes at the end of each chapter of the book.

Clearly, historically we have to start already with medieval times, namely with Gothic church windows. But coming to mathematics, we first have to mention three famous names. Namely, Euler (cf. [323]), Minkowski (see [837] and the interesting second appendix of [835]), and Lebesgue (see [707]) looked already explicitly at the phenomenon of noncircular constant width sets. More precisely, Euler was most likely the first who explicitly looked for mathematical properties of curves of constant width and called them orbiforms (see also the historical study [1199]). He was interested in the construction of closed convex curves (called catoptrices) having the property that a ray, starting from an interior point of the curve, reaches this point again after two reflections at the curve (the ellipse is a well-known example). He started with so-called triangular curves having three concave sides and three cusps, and he showed that the evolvent of them is a curve of constant width, which then was used to construct a catoptrix. In large parts, Minkowski created the fundaments of convexity and, via the geometry of numbers, also of parts of discrete geometry. Many basic notions (like support function, see, e.g., [836]) are due to him, and in his early works one finds also investigations of bodies of constant width (see [837] and the second appendix of [835]). Lebesgue [707] proved that the Reuleaux triangle has the smallest area among all curves of the same constant width (the circle clearly yields the other extremum), and one year later this was independently reproved by Blaschke [131]. Here we also mention that the astronomer and mathematician Barbier proved already in 1860 that all planar sets of fixed constant width h have the same perimeter, namely $\pi h$, see [75]. Another historical milestone is the contribution of Reuleaux, a German professor in mechanical engineering. In section 22 of his classical book [969] (see also the German original [968]) on the theory of machines he described the curves called after him, and their properties were developed in later sections of that interesting monograph. More historical background regarding Reuleaux and other contributors to the field of kinematics in engineering is given in the nice book [854]. We mention here also the interesting paper [648] about Burmester, distinguished professor of descriptive geometry and kinematics at the University of Munich. He was responsible for much of the development of scientific mechanical engineering in the nineteenth century, and the article [648] interestingly illuminates also the rivalry between Burmester and Reuleaux.

We continue now with mentioning books and expository articles in which constant width sets are discussed. The scheme of our discussion is (simply following the occurrence in the literature) structured by the sequence classical geometry, recreational mathematics, differential geometry, and convexity.

Thus, we note first that, with their beauty and impressive variety of shapes, curves and bodies of constant width are well established in classical geometry, occurring in correspondingly famous monographs like in § 32 of [540] and in subsection 12.10.5 of [100]. They are also discussed in history of mathematics (see, e.g., § 7.1 of [1053]) and occur as interesting mathematical models in many collections and publications, cf. [1029],  [818],  [274],  [362], and chapter 2 of [143] (the latter containing also a broad mathematical discussion).

Books close to recreational mathematics or education in mathematics, suitable for students and also for interested non-mathematicians, are [550] and [26], not to forget the treasure of related publications of Martin Gardner appearing often as sections in his popular, nice books (see, e.g., [398]). Typical goals in such books are the repeatedly occurring descriptions of nice properties of Reuleaux polygons, constructions also of general constant width sets and their parallel curves, and applications (like rotors). Such a detailed discussion of the construction of plane curves of constant width is, for example, given in the nice elementary book [950], pp. 107–123; discovering mathematical facts is presented there in the form of discussions between Salviati, Sagredo, and Simplicio in the style of Galileo’s famous book. Here we mention also the paper [537], in which the authors found a way to define and investigate sets of constant width on a chessboard with n rows and n columns.

From the viewpoint of differential geometry and classical curve theory, plane and spatial curves or surfaces of constant width are interesting geometrical objects, see Chapter 11 and also [731], to give also an early comprehensive reference. Unfortunately, we are not aware of a monograph discussing in a systematic way properties of surfaces of constant width in the spirit of differential geometry (but we refer, however, to Chapters 11 and 17 here). Only the planar case is handled in several classical books. With their relation also to Zindler curves, plane curves of constant width are presented in the first chapter of [1103], and in connection with isoperimetric properties and vertex theorems they occur in Chapter 3 of [830]. Extremal properties of plane curves of constant width are discussed in Chapter 2 of [561], and two further books to be mentioned here are [1147], written as a classical course in differential geometry and studying also curves of constant width in the first chapter, and [18], where the concept of constant width can be found in the fifth chapter.

We come now to books discussing constant width sets in the spirit of convexity. Early ones are [132] (see appendices III and IV there, where also a body of constant brightness is presented), [958, § 20b], and [160, § 15], followed by the monograph [1204]. Three problem books, in which constant width sets and closely related topics occur directly or indirectly several times, are [311],  [272], and [635]. Textbooks in convexity, suitable for teaching undergraduates or graduates and containing sections about constant width sets are [312],  [1140],  [98] (containing also a chapter on Minkowski geometry), [737],  [706],  [618],  [1134],  [152],  [654],  [1163],  [850] and [719] (the latter together with [720]). Concrete subjects discussed therein are, e.g., constructions, the Blaschke–Lebesgue theorem, Reuleaux polygons, completeness, related notions (e.g., equichordality), Barbier’s theorem, the incircle–circumcircle relation, double normals, and sometimes also 3-dimensional analogues. There are also books from neighboring fields, where constant width sets are shortly presented. As example we mention the field of geometric probability, with the famous monograph [1021] (see Section 4 in Chapter 1 there).

Monographs being closer to recent research on bodies of constant width and related notions are [1039, Chapter 3], [464, Chapter 5], [401, Chapters 3, 4, 6, 7, and 9], and [1124, Chapter 4]. Subjects and applications discussed there in a deeper way are projection and section functions, rotors, completeness, geometric inequalities, related notions like constant brightness or equichordality, constant width in Minkowski spaces, and others. One should also underline once more the historical importance of the book [160, § 15] regarding research on constant width sets in the first half of the twentieth century.

Finally we mention surveys and expository articles on sets of constant width and their main aspects. The most comprehensive representation of the field with a large variety of references (after [160] and until 1983) is the basic survey [238], covering practically all aspects of constant width sets. Ordered with the same arrangement of aspects, this survey was updated by [1192](1991) and [527] (1993), and in 2004 (again with the same arrangement of aspects) an analogous survey on constant width sets in Minkowski spaces followed, see Section 2 of [793]. A wonderful expository paper on curves and surfaces of constant width is [970] which, unfortunately, was not published until now. Many aspects are discussed there, and various new results are also derived. Also in the exposition [85] curves and surfaces of constant width in two and three dimensions are discussed. Two older surveys on constant width sets are [598] and [99]. In the first one, basic properties (mainly in the spirit of our Chapter 3) are reproved, and the second one gives an update of § 15 of [160]; basic properties of planar constant width sets and inequalities for them are derived, and also curves of constant width in space and in Minkowski geometry are presented. Two further papers, presenting known results in a really nice way, are [137] and [363]. For example, the elementary paper [363] is based on the characterization of constant width sets in terms of Fourier series, setting the used context at a level suitable for undergraduates; it is also presented how to construct sets of constant width from midpoint curves. Two older papers, in which known results on curves and surfaces of constant width are presented in a new analytic way (e.g., via support functions) and their mechanical applicability regarding the transformation of alternating movements is shown, are [1127] and [1128]. The authors of [161] and [1161] had certainly similar aims; parts of their contributions are also close to mathematical education. The survey [612] describes (with some history and nice figures) the state of the art regarding the conjecture that Meissner’s bodies are of minimal volume among all 3-dimensional bodies of the same constant width. The paper gives a good overview to the problem, is at the same time readable for experts and non-experts, and collects known results supporting the conjecture (here we mention also the problem paper [609] and [272, A 22]). Two further elementary surveys are [1130] and [85]. The first one refers only to basic properties in the planar case, and in the second also properties of 3-dimensional constant width bodies are derived, where the support function is the main tool used there. The problem collection [1106] and the surveys [302] and [722] refer to problems from convexity and contain parts discussing nicely constant width sets, see also [823] and [825].