**Reuleaux Triangles**

The most popular noncircular set of constant width in the plane is the *Reuleaux triangle*, whose first mechanical application is ascribed by Reuleaux himself (see [968], § 155) to Hornblower, the inventor of the compound steam engine. We refer also to our Chapter 18 for many related results.

Replacing boundary parts of planar constant width sets by arcs of circles, new sets of constant width may be obtained. This was used in [997] to show that if there is a set of maximal constant width whose interior misses a locally finite family of convex sets, then it is a Reuleaux polygon with at least one contact point on each edge. Analogously for lattices, a unique maximal set exists and forms a Reuleaux triangle. Schmitz [1030] showed that the total length of two orthogonal intersecting chords of a Reuleaux triangle of width 1 is larger than 1. This observation motivated Martini and Makai to conjecture and prove Theorem 4.4.2 (see [753]). For the corresponding investigations in normed planes, see [22, 25], and the notes to Chapter 10. Stability results on the Hausdorff distance of sets of constant width to Reuleaux triangles (and circles) are derived in [458], and Weissbach [1195] obtained the following characterization of Reuleaux triangles: A convex set *K* is called a covering set of a class *Q* of planar sets if any member of *Q* can be covered by a congruent copy of *K*; *K* is particularly a minimal covering set if it cannot be replaced by a proper subset. It is shown in [1195] that there are minimal covering sets for the class of all planar sets of constant width, except for Reuleaux triangles. Chakerian [234] proved that if a convex body in the plane can be covered by a translate of a Reuleaux triangle, then it can be covered by a translate of any set of the same constant width, see Theorem 15.1.2. In [1094], it was shown that the sum of the diameters of three sets, each of diameter smaller than *h*, which together cover a set of constant width *h*, is greater than 2*h*. The main proof part depends on the special case of a Reuleaux triangle; namely, the perimeter of a triangle whose vertices belong to distinct arcs of a Reuleaux triangle of width *h* must be larger than 2*h*. For a planar, \(C^2\)-smooth set \(\Phi \) of constant width 1, say, and the subset of \(\Phi \) consisting of all points lying on at least three diameters of \(\Phi \), Makeev [759] showed that the area of this subset has an upper bound which is attained if and only if \(\Phi \) is a Reuleaux triangle. In [516], the so-called *m*-diameter of a set *S* is defined as the supremum of the geometric mean of all Euclidean distances among *m* points from *S* (thus, \(m=2\) yields the ordinary diameter of *S*). It is proved that for constant width sets in the plane, circular disks have the smallest and Reuleaux triangles the largest 3-diameter. The Reuleaux triangle also occurs in extremum problems from distance geometry in metric spaces, see [936]. A type of “generalized Reuleaux triangle” is introduced in [1202].

**Reuleaux Polygons and Related Notions**

*Reuleaux polygons* are defined as those planar sets of constant width *h* whose boundary consists of a finite, odd number of circular arcs of radius *h*; their centers are called the vertices of the Reuleaux polygon, and they also form the vertex sets of a star polygon, whose edges are the extremal segments of the diameter graph of the considered set (see [160], pp. 130–131). Reuleaux polygons were introduced by Reuleaux himself (cf. [968] and also [1028]). Blaschke ([130] and [131]) showed that arbitrary sets of constant width *h* can be approximated arbitrarily closely by Reuleaux polygons of width *h*, see also [192], and for a quantitative version relating the Hausdorff distance to the odd number of sides, [603]. We also refer to Section 8.1. In [417], regular Reuleaux polygons are characterized among all constant width curves by an extremal property of the perimeter of circumscribed equi-angular polygons. On the other hand, Zalgaller [1210] proved extremal properties of certain convex *m*-gons with the help of Reulaux polygons. In [868], Reuleaux polygons are discussed in connection with isodiametric-like problems for convex polygons, and in [672] a simple characterization of those geometric graphs is given which are diameter graphs of the vertex set of Reuleaux polygons. From this, a *linear-time construction of Reuleaux polygons* was obtained, clarifying the basis for traditional constructions of Reuleaux polygons involving the set of diameters. We mention here some further related algorithmical results. Continuing [227], the authors of [228] presented two linear-time ruler and compass constructions of figures of constant width *h* circumscribed about a given simple polygon whose diameter is *h*. Their approach is closely related to algorithmical work of Hershberger and Suri on the notion of circular hull, see [538]. Since circular intersections, circular hulls and their higher dimensional analogues carry over to normed planes and spaces and are important there (see e.g., the subsection on intersection properties in the Notes of Chapter 10), we mention here also the papers [773], in which the results from [538] are extended to normed planes, and [775], where the same is done also for nonsymmetric convex distance functions.

A geometric graph all whose edges intersect is an intersector. In [669], close connections between intersectors (with segments of equal lengths) and curves of constant width are discussed, in particular also referring to Reuleaux polygons. Sgheri [1058] studied the deformation of regular Reuleaux polygons into non-regular ones. In [317], it is shown that among the largest equilateral triangles, which may be inscribed in different curves of constant width, that one inscribed in a “Reuleaux pentagon” is the smallest. In [331], a point *x* of a set of diameter 1 is called a “singularity” if there exist two different points in this set whose distance to *x* equals 1. By a clever construction, the author showed that every compact subset *S* of \({\mathbb E}^n\) with diameter \(\le 1\) can be covered by a set *Y* of constant width 1 such that *S* and *Y* have the same set of singularities and that these are of the same dimension. By a related construction, in [927] constant width sets with a given number of singularities are generated. Hammer [502] presented a generalization of Reuleaux polygons for normed planes whose Euclidean subcase yields constant width sets formed by a finite number of circular arcs having different radii, see also [958], p. 167, and [398]. The study of constant width curves representable as involutes of hypocycloids or further types of curves goes even back to Euler, see [160], pp. 131–132, [197, 408, 674], and [1144] for related results, and see also Section 5.3.4. Silverman [1064] investigated indecomposability (in the sense of Minkowski addition) for certain classes of convex sets, and from her results it follows that Reuleaux polygons are indecomposable within the family of constant width sets in the plane, see also [603]. In many situations, Reuleaux polygons are also popular geometric objects for demonstrating geometric facts in a suggestive way. For example, in [1126] they are used as examples for observations around Holditch’s theorem. They play also a role for number-theoretic Favard-type problems concerning algebraic integers (see [680]): Depending on the notion of transfinite diameter and its relation to that of usual diameter, certain subsets of the complex plane suggest some interesting conjectures of a purely geometric nature, concerning also the transfinite diameter of Reuleaux polygons. For related results, we refer also to the papers [678, 679], and [681]. The concept of *m*-diameter (see above, where we refer to Reuleaux triangles) yields also results on Reuleaux pentagons. Namely, in [570] it is proved that the Reuleaux pentagons have the largest 5-diameter among all sets of given constant width.

**Constructions of Curves of Constant Width**

Let *U* be an open set in the plane, and let \(\alpha \) be a real number, \(0< \alpha \le \pi \). Saroldi [1023] introduced the angle property of opening \(\alpha \) for *U*: this property is fulfilled if every point on the boundary of *U* is the vertex of an angle of measure \(\alpha \) whose interior does not intersect *U*. Thus, the angle property is a natural generalization of convexity, and it is equivalent to usual convexity when \(\alpha = \pi \) (if *U* is connected). The main result in [1023] is that if *U* is such a set of diameter *h*, then the perimeter of *U* is less than or equal to \(\pi h/\sin 2(\frac{\alpha }{2})\). This is the best upper bound when \(\frac{\pi }{2} \le \alpha \le \pi \). For \(\alpha = \pi \), any convex region of constant width *h* will have perimeter \(\pi h\). If \(\alpha < \pi \), then one cannot achieve the upper bound for the perimeter, but one can approach it arbitrarily closely (at least for \(\alpha \ge \frac{\pi }{2}\)) by a construction whose starting point is a region *U* of constant width *h*. In [1144], the notion of curve of constant width is extended in a constructive way, namely to all closed curves with multiple points such that each normal is a double normal, being normal to the curve at two points of it having constant distance. The author considers certain curvilinear polygons with an odd number of sides, with a cusp of the first type at each vertex, but such that each side is a simple arc without further singularities. Then multiple curves of constant width are constructed by describing an involute of such a polygon, and relations involving the total curvatures of both families of curves are given, see also our Chapter 11. In [408], constructions of constant width curves as involutes of certain curvilinear “star” arrangements of convex arcs (continuing [160], p. 132) are discussed. Geometrical constructions of curves of constant width are also used in computational geometry to derive time complexities of other problems, like polygon simplicity testing etc., see, e.g., [578]. Constructive approaches to rational ovals and rosettes of constant width formed by piecewise rational Pythagorean–Hodograph curves are presented in [8], see again also Chapter 11. A nice construction of a curve of constant width consisting of four arcs is given in [1203], where an isosceles trapezoid is the starting point and the two obtainable borderline cases are the disk and the Reuleaux triangle. Similarly, [786] showed how a planar constant width curve can be continuously constructed from a Reuleaux triangle while preserving the constant width property in all intermediate steps of this process. In [389], a necessary and sufficient condition for a convex arc of class \(C^2\) is given to be extendable to a closed convex curve of constant width.

Based on Eggleston [316], where it is proved that a compact set *X* of diameter *h* has constant width *h* iff *X* is equal to \(B(X) = \cap \{B(x, h):x \in X\}\), Sallee [998] used the spherical intersection property to construct sets *K*(*Y*, *D*) of constant width *h* containing a prescribed set *Y* of diameter *h* and determined by a sequence \(D=\{x_n\}\) which is dense in \({\mathbb E}^n\). Defining \(F_0 = B(Y)\) and \(F_n = F_{n-1} \bigcap B(x_n, h)\) or \(F_{n-1}\) as \(x_n \in F_{n-1}\) or \(x_n \notin F_{n-1}\) and then setting \(K(Y, D)=\bigcap \{F_n : n = 0,1,2,\dots \})\), he employs this construction (which was already used in [997] and [999] in a modified way) to define Reuleaux polytopes and to show that for the 3-dimensional case this class is dense in the family of constant width bodies regarding the Hausdorff metric, see Sections 7.2 and 10.3.

In view of completing convex bodies (see Chapter 7), Schulte [1048] presented the following nice construction: For a convex body *K* of diameter 2 in \({\mathbb E}^n\), there exists a body \(\Phi \) of constant width 2 containing *K* such that every symmetry of *K* is one of \(\Phi \) and every singular boundary point of \(\Phi \) is a boundary point of *K* for which the set of antipodes in \(\Phi \) is the convex hull of the antipodes in *K*. (As a consequence, the author proved Borsuk’s conjecture for convex bodies having no point as endpoint of more than one diameter.) In [1049] and [1050], these investigations are continued, see also Section 7.5 and the Notes to our Chapter 7. Bavaud [81] provided an exposition of the properties of the adjoint transform, associating to a set the intersection of all disks of given radius centered at the set. The relationships between this transform, the double adjoint transform, basic notions from convexity (like also completion of sets and the concept of constant width) are studied, see also Section 7.3. Further constructions of orbiforms from polygons are investigated in [228].

**Constructions in Higher Dimensions**

A century ago, Meissner introduced special convex bodies of constant width which, constructed from a regular tetrahedron, are now named after him. It is a well known and still unsolved conjecture that these *Meissner bodies* have the smallest volume among all 3-dimensional convex bodies of the same constant width, see also Sections 8.3 and 14.2. A well-written and nicely illustrated survey describing the Meissner bodies, presenting their history, and explaining some recent observations regarding that extremal problem is [612]. A short biography of Ernst Meissner is also there. In [545], it is asked whether it is possible to confirm this conjecture with the help of computers. However, besides Meissner bodies the “easiest” type of concrete 3-dimensional bodies of constant width are simply obtainable via rotation of axially symmetric 2-dimensional ones, taking an axis of symmetry as rotation axis. In [1136], volume and surface area of the body obtained in such a way from the Reuleaux triangle are computed and compared with the analogous quantities of Meissner bodies. The starting point of the Meissner construction, the intersection of four congruent balls called Reuleaux tetrahedron (which is clearly not of constant width), shows the necessity of rounding off curvilinear edges of such ball polytopes, see Section 8.2. Weissbach [1191] investigated the *n*-dimensional analogue, computing its diameter. This process of rounding off curvilinear edges was described by many authors, see, e.g., [160], pp. 135–136, [1204], § 7, and [151], pp. 378–379. Besides the tetrahedral case, this was studied by Gray [448] also for other types of pyramids. An older related paper is [192], where so-called “elementary bodies” in \({\mathbb E}^3\) as analogues of Reuleaux polygons are constructed, namely as intersections of balls of radius *h*, whose centers were chosen at the members of a finite point set or at points of circular arcs of radius *h*. It turns out that bodies of this type can approximate any 3-dimensional constant width body as precise as one wishes. These investigations were continued by Sallee [998] with another construction, but yielding the same approximation property.

Note that constructions of this type, yielding Reuleaux polytopes, are also discussed in [853], see also Chapter 6. In [677], various characteristic properties of constant width bodies in \({\mathbb E}^n\) are derived, and a construction of several types of *n*-dimensional constant width bodies is presented, which has a given (\(n-1\))-dimensional constant width body as an orthogonal projection (one of the bodies obtained in this way is a classical Meissner body). For a similar construction see Section 8.4.3. However, in contrast to the constructions in [853], which give rise to finite procedures (see Sections 8.4.1 and 8.4.2), the procedures in [677] and Section 8.4.3 are not finite because they still need to intersect an infinite collection of balls. Danzer [276] constructed a 3-dimensional body of constant width *h* such that the minimum width of each of its 2-dimensional sections are smaller than *h*, see Section 9.2. It is also natural to ask for symmetry properties of bodies of constant width in higher dimensions. Usually, explicit higher dimensional constant width bodies presented until now have some symmetry properties, like rotational or tetrahedral symmetry. In [350], it is shown that there are analytic hypersurfaces of constant width in \({\mathbb E}^n\) whose symmetry group is trivial, and that, on the other hand, there are also *n*-dimensional constant width bodies with analytic boundary, having the symmetry group of the regular *n*-simplex. (A proof gap in [350] can be filled via constructions presented in [982] and [1048].) For \(n=3\) it is also proved which subgroups of the rotational groups occur as symmetry groups of surfaces of constant width. Related to this and motivated by Borsuk’s problem, Rogers [982] proved that for any set *S* of diameter 1 and invariant under a group *G* of congruences, leaving a point invariant, there is a set of constant width 1 containing *S* and being invariant under *G*. Surprisingly, it is shown that in \({\mathbb E}^8\) a set of constant width 1, invariant under the symmetry group of a regular 8-simplex, exists that has no cover, invariant under this group, by nine sets of diameter smaller than 1. Coming back to Meissner bodies, we mention that they play also an important role in view of various geometric inequalities; see e.g., [591] and our Chapter 14.

**Analytic Representations**

Explicit representations of planar constant width sets in terms of polar coordinates are presented by Hammer [502], Kearsley [614], and Tennison [1122]; in the latter case also an analytic representation of smooth curves of constant width in terms of their curvature is given. Rabinowitz [956] applied the classical parametric representation of a plane convex curve in terms of its support function to derive curves of constant width. Using Mathematica, one of the simpler examples is converted to a polynomial equation for the curve, see Section 5.3.3. The question for the lowest degree polynomial whose graph is a noncircular curve of constant width is raised. Expressing it as an algebraic curve, isolated points not from the original curve are obtained; in [914] it is shown how to avoid this problem, i.e., how to construct a constant width curve in this way that has no isolated points. The authors of [86] present a complete analytic parametrization of 3-dimensional constant width bodies. For parametrizations of 3-dimensional constant width bodies see also Sections 8.5, 5.3.4 and 11.1.

**Approximation Results**

Recall that as starting point for the discussion of approximation results on constant width sets, already Blaschke ([130] and [131]) showed that they can be approximated by Reuleaux polygons in an arbitrarily good manner. Some other results on approximations are also contained in the discussion of 3-dimensional constructions, in particular of Meissner bodies (see Section 8.3). Bodies of constant width occur in the framework of approximation results in the surveys [470, subsection 2.4] and [476, Section 7]. Schneider [1038] showed that each convex body in \({\mathbb E}^n\) can be approximated arbitrarily closely by convex bodies having an algebraic support function and everywhere positive radii of curvature, and the approximating bodies can be chosen in such a way that they have at least the same group of symmetries as the approximated body. Special care is taken for the concept of constant width, and it is shown that for constant width bodies one may choose approximating bodies of constant width in this way, see also Theorem 8.5.1. These results nicely continue related investigations and results from [1118] and [1164]. Aumann [50] gave approximations of constant width sets in the plane with the help of Bézier splines, and he also proposed algorithms for determining quantities like their width, inradius, and circumradius. In [393], the evolution of strictly convex curves with a representation under some special flow (different to the known and usual curve shortening flow) is studied. With demands like smoothness and curvature conditions on the initial curve, this flow tends to curves of constant width, and thus also leaves curves of constant width invariant. Kharazishvili [620] showed that if a plane curve of constant width is sufficiently smooth, then it can be approximated by algebraic curves of constant width. Having uniform approximation of periodic functions in mind, the authors of [727] studied the approximation of convex curves by piecewise circular curves, discussing also constant width curves. Falconer [331] proved that any *n*-dimensional body of constant width *h* can be approximated by another body of the same constant width *h* having a dense set of singularities (= points of that body having at least two other points from it at distance *h*). The authors of [1097] derived results on discontinuities of measures approximating the 1-dimensional Hausdorff measure for planar sets of constant width; they also gave some sufficient condition for having no such discontinuities. The necessity of these conditions was then confirmed in [1092]. See also [1091, 1093], and [1094], where propositions about these continuity results on approximation measures of constant width sets are added. Further similar results in this direction are presented in [1095] and [1096]. For a given integer \(m \ge n+1\), in [451] estimates on the minimal width of convex polytopes are given, where these polytopes have at most *m* vertices and are inscribed to bodies of given constant width. In special cases, even explicit values are obtained. Via the spherical intersection property, the notion of spindle convexity is close to that of constant width, see our Chapter 6. Two papers on spindle convexity are closely related to the approximation concepts discussed here. A compact spindle-convex set in \({\mathbb E}^n\) is the intersection of a finite family of closed unit balls, and a convex disk-polygon is the 2-dimensional variant thereof. In [372], the authors study the approximation of spindle convex sets with twice continuously differentiable boundaries by circum- and inscribed convex disk-polygons with at most *m* sides regarding Hausdorff metric, area deviation, and perimeter deviation. And in [370] famous results of Rényi and Sulanke are carried over from usual convexity to spindle-convex sets: Given a compact, spindle-convex set in the plane satisfying certain boundary conditions (e.g., smoothness), the authors take *m* independent uniform random points in this set and derive asymptotic formulae for the mean number of vertices, area, and perimeter deviation regarding their spindle-convex hull. The papers [192] and [998] contain also related approximation results.