**Line Families**

Line families related to sets of constant width occur in different forms combined with the notions of *outwardly simple line families (or systems of externally simple lines), pedal functions* and *pedal curves, envelopes, evolutes/involutes, hedgehogs*, *evolutoids*, and *ruled surfaces*. In the following we want to collect references in which these notions are applied to sets of constant width, or which are combined with constant width sets. Since almost all these concepts are closely related to differential geometry, the reader is also referred to our Chapter 11.

Hammer and Sobczyk (see [511] and [512]) studied the configurations of line families coming from diametrical chords of planar constant width sets. They called a family of straight lines in the plane an *outwardly simple line family* (or *system of externally simple lines*, see Section 5.3), if it covers the exterior of some circle simply (see also [220] for further results on such line families). Additionally they proved the fact that all orthogonal trajectories of an outwardly simple line family having a point outside of a sufficiently large circle are smooth constant width curves. Moreover, all such curves can be obtained in this way. Hammer [509] summarized this work nicely, referring also to extensions for normed planes. The papers [506], [507], [508], and [1072] are also related, containing negative results for \(n > 2\). In [508], outwardly simple line families are explicitly used as a tool to get analytic expressions for all types of constant width curves; the author starts with the line families in the smooth case and then describes the eventually occurring corners via some limiting process. In this way one can also obtain nice characterization theorems, see [541], [104], [510], and [829]: e.g., if each diametral chord of a constant width curve bisects its circumference (its area) then this curve is a circle, see Corollary 5.1.4. Since diametral chords lie on double normals, the following results should also be mentioned here. Kuiper [663] has shown that the set of lengths of double normals of a convex body in \({\mathbb E}^n\) is of measure zero if \(n < 4\), while for all higher dimensions there is a convex body not of constant width with the property that the range of its width function coincides with the set of lengths of its double normals. Heil (see [525] and [526]) proved that a 3-dimensional body of constant width either contains a point belonging to infinitely many normals, or an open set of points through each of which at least 10 normals pass. Let *m*(*K*, *p*) denote the number of normals of a convex body *K* passing through \(p \in K\), and *m*(*K*) be the average of *m*(*K*, *p*) with *p* running through *K*. Chakerian [236] could provide sharp bounds on *m*(*K*) for planar constant width sets; the lower bound characterizes the circle, the upper bound the Reuleaux triangle. Also in [301] upper bounds for *m*(*K*) are proved, and related results on *n*-dimensional bodies of constant width are derived.

It is clear that the concept of pedal function, introduced in Section 5.1, and the headline of this chapter are closely related to the notion of pedal curve (a special case of envelopes). Namely, the *pedal curve* of a plane curve *C* with respect to a fixed point *y* is the locus of all points *x*, such that the line spanned by *x* and *y* is perpendicular to the support line of *C* passing through *x*. Considering support hyperplanes instead of lines, this notion can be carried over to hypersurfaces in \({\mathbb E}^n\), continuing the considerations from the booklet [597] on parallel curves of constant width curves. Kelly [617] showed the following: If a closed convex plane curve has constant width, then its pedal curve with respect to an arbitrary point *y* has *y* as an equichordal point (i.e., all chords of it passing through *y* have the same length). Conversely, given a curve, star shaped with respect to *y* and having *y* as an equichordal point, its “negative pedal curve” (easily obtained as an envelope of straight lines by reversing the procedure) has constant width provided it is convex. In the same manner Kelly also considered inner parallel curves of curves of constant width. Given a plane convex body of area *F* with boundary curve *C* of class \(C^2\) and fixed length, let *A* denote the area enclosed by the pedal curve of *C* with respect to the Steiner point of the given convex body. As shown in [273], the isoperimetric deficit of that body has the lower bound \(3 \pi (A-F)\). Based on this, an inequality from Groemer’s book (see Theorem 4.3.1 of [464]) is improved, holding for the special case that the given set is of constant width.

The concept of pedal curves and Kelly’s result have natural analogues in higher dimensions. For example, a *hedgehog* is the envelope of a family of hyperplanes \(\langle x, z\rangle =h(z)\), where *h* is a function of \(z \in S^{n-1}\), the unit sphere. It is a hypersurface \(H_h=x_h(S^{n-1}) \subset {\mathbb E}^n\), and for a point \(x_h(z)\) belonging to the smooth part of \(H_h\) the unit normal vector is *z*. The mapping \(x_h\) can be interpreted as the inverse to the Gauss map of \(H_h\). In [778], a higher dimensional analogue of Kelly’s result above is derived, referring via antipedal hypersurfaces to hedgehogs of constant width or ensuring (under smoothness assumptions) that the pedal hypersurface of a given hedgehog of constant width is convex. In [777], and [780] related characterizations of spheres among constant width sets are given. Further results on hedgehogs referring also to the concept of constant width can be found in [776], [779], [780], [781], [782], and [1042]. For example, there is a 6-normals theorem for hedgehogs of constant width (see [779]), and [782] refers to evolutes within this framework (see also below).

Now we come to the concepts of *evolutes* and *involutes*. Already Euler [323] studied orbiforms as involutes of three-cuspid curves, and related representations can be found on pp. 313–316 of the basic book [731], see also Section 5.3.4. Essential in this direction are the papers [598] and [1028], and in the monograph [160] constant width curves are discussed as involutes of certain “*k*-cuspid” curves (see § 65 there). Based on this, also Gere and Zupnick [408] constructed constant width curves as involutes of such “star curves” consisting of an odd number of convex pieces of total curvature \(\pi \). They derived necessary and sufficient conditions for such star curves to play that role in terms of inequalities satisfied by their “side-lengths”. Kurbanov [674] studied special curves of constant width being involutes of the hypocycloids with an odd number of cusps, and further related papers are [197], [1143], and [1144]. In [1144], the definition of constant width curves is generalized to include all closed curves with multiple points such that each normal is a normal to two points of the curve at constant distance. The author considered certain curvilinear polygons with an odd number of sides and constructed from them such multiple curves of constant width using involutes of these curvilinear polygons. For analogous considerations in normed planes we refer to Petty [929]. To verify an important result on the approximation of constant width curves by \(C^{\infty }\)-ovals of constant width, Tanno [1118] used and modified the related evolute–involute relation. A related higher dimensional result for rotational hypersurfaces of constant width is also given, and a generalization of the planar approximation result from [1118] is given in [1171]. Ostrowski [904] studied so-called finite ovals as curves with continuous, never vanishing curvature and an even number of vertices, and he gave also a precise description of their evolutes. He applied then these considerations to obtain extensions of some results on curves of constant width, due to Schilling [1028], under weaker assumptions. The authors of [446] studied evolutes of smooth closed convex curves in the plane, on the (generic) assumption that the evolute has only ordinary cusps and transverse crossings as singularities. In case that the evolute has exactly (the minimum number of) three cusps, the considered curve is of constant width, and the evolute has no self-crossings. Let a constant width curve be of class \(C^4_+\). It is known that such a curve has at least 6 vertices, and its interior contains either a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals. If all its vertices are nondegenerate, then such a curve has exactly 6 vertices if its evolute is the boundary of a topological disk through each interior point of which pass at least 6 normals, see [782]. We mention here also the papers [1144], [1156], and [1157], in which properties of plane curves with possible self-intersections and a kind of double-normal property are investigated, e.g., also when they are closed.

Related to the concept of evolutes, an *evolutoid* is a curve obtained as the envelope of lines making a fixed angle with the normal line at every point of a given curve. In [581], it is shown that a convex curve is of constant width if its evolutoid for a fixed angle is of constant width. These investigations have their continuation in [4], where also relations to the famous homothetic floating body problem are discussed and it is proved that a curve and any of its evolutoids have the same Steiner point. Moreover, relations between evolutoids and constant angle caustics are also shown, and also curves of constant width play a role in this paper. The paper [872] addresses secantoptics of a closed convex curve (i.e., isoptics of its evolutoids). It is proved that for constant width curves the distance between opposite points of the secantoptics remains constant, and that, conversely, under certain conditions this property characterizes curves of constant width. Isoptics of constant width curves are studied in [256], and for the more general rosettes of constant width we refer to [828], see also the first part of the notes to Chapter 11.

Furthermore, *ruled surfaces of constant width* should be mentioned here since they are also related to line families. For example, Nádeník [884] investigated surfaces generated from the normal hyperplanes of a closed analytic curve as follows. Considering a closed convex (\(n-2\))-dimensional surface *W* in such a normal hyperplane, one can study the cylindrical surfaces having *W* as right section. Assuming that the envelope of these cylindrical surfaces is free of singularities and that *W* is of constant width, typical properties of constant width surfaces are proved. Wegner [1173] defined cylinders of constant width in 3-space as complete \(C^{\infty }\)-surfaces *V* of the topological type of a cylinder such that, for each \(p \in V\), the normal line *l* at *p* intersects *V* again in a point \(p'\), where *l* is also the normal line of *V* at \(p'\). For example, it is proved how an important subclass of these surfaces can be generated by moving a planar curve of constant width along an appropriate space curve. And if *V* is a cylinder of constant width fibered by a one-parameter family of plane curves of constant width, then the preceding construction can also yield *V*. With the help of results from global kinematics in 3-space, Pottmann [952] constructed ruled surfaces, which generalize curves of constant width and Zindler curves, see Section 5.4.1. In [554], skew ruled surfaces in 3-space are considered whose generators have a constant angle with respect to a fixed 2-plane, and in which constant width projections of these surfaces suitably occur. For such surfaces a six-vertex theorem and some analogue of Barbier’s theorem are proved. Also in the survey [647] ruled surfaces of constant width are discussed (see § 5.3 there).

**Floating and Related Billiard Problems**

Floating problems and billiard problems related to them are for example discussed in the problem book [272], more precisely in the sections A 4 and A 6 there. Further references on billiards are collected in [497] (for the polygonal case) and in [1137] (for the smooth case). The paper [352] refers to rigid bodies on liquid surfaces subject to capillary (and not gravitational) forces. In equilibrium, the liquid surface must have constant mean curvature (law of Laplace), and at the “contact line”, the contact angle between it and the boundary of the floating body has to take a given value (Young’s law). The author shows that if this contact angle is \(\frac{\pi }{2}\) in three dimensions, and the floating body is smooth and strictly convex, then the spherical shape is obtained. But in two dimensions and again for contact angle \(\frac{\pi }{2}\), any smooth constant width set admits a neutral equilibrium for any possible orientation. The notion of constant width is only relevant for this isolated case \(\frac{\pi }{2}\), and the used construction achieves its goal as the orthogonal trajectory of a line family, without using the constant length of the line segments cut out of the arising geometric figure. It is nice that the regular analytic curve of Rabinowitz [956] is also used in [352], see Section 5.3.3.

In the related note [353] the connection of such floating body problems and billiard caustics is discussed, citing also the earlier works [351] and [498]. Gutkin [499] presented a detailed solution of problems on capillary floating in terms of billiard problems. Also here, in a special case, curves of constant width are essential. Quantum systems with time-reversal invariance and classically chaotic dynamics have, by common assumption, energy spectra distributed according to the Gaussian orthogonal ensemble type of statistics. Gutkin [496] introduced a class of smooth convex billiards of constant width whose dynamics are time reversible and “almost” chaotic. In [68], it is shown that any minimal closed billiard trajectory in a planar body of constant width is 2-periodic, see Theorem 17.2.3. The proofs are based on results from [108], with analogous results referring to disk polygons (i.e., 2-dimensional ball polytopes).

We now come to caustics in relation to constant width (see again the notes in Chapter 17). In the beautiful paper [640] the geometry of caustics of plane billiard tables of constant width is studied; various nice figures are presented there. Using results on the so-called Wigner caustic, in [1225] an improved version of the isoperimetric inequality is given, yielding also a characterization of constant width curves.

**The Floating Body Problem**

Problem 19 of the *Scottish Book* [807] is still open in dimension three and only few special cases have been solved. In three dimensions, in the density limit \(\rho \rightarrow 0\) or 1, there are no solutions other than the sphere [842]. Independently, Falconer and Schneider proved that there are no nontrivial solutions among star-shaped objects with central symmetry for density \(\rho =\frac{1}{2}\) (see [332] and [1032]). Falconer’s proof, using spherical harmonics, will be presented in Section 13.3.1. On the other hand, F. Wenger has proposed a perturbation expansion scheme starting from the sphere for objects with central symmetry and \(\rho \not =\frac{1}{2}\) (cf. [1183]), as well as for bodies with arbitrary shape and \(\rho =\frac{1}{2}\), see [1184]. His results point out the existence of many nontrivial solutions in these wider classes of shapes, even though the proofs are incomplete in that the convergence of the perturbation series has not been examined. Furthermore, no attempt to construct actual solutions of the problem in dimension three has been reported.

Regarding the 2-dimensional floating body problem, when the density is not necessarily one half, F. Wenger gave a solution, showing the existence of noncircular figures that float in equilibrium in every position. If a figure *F* of density \(\rho \) floats in equilibrium in every position, then we know that the water surface divides the boundary of *F* in a constant ratio, say \(\sigma / 1-\sigma \). We call \(\sigma \) the perimetral density of *F*. In [1181], F. Wenger was able to obtain noncircular solutions for \(\rho \not =\frac{1}{2}\not =\sigma \), by a perturbative expansion around the circular solution. These figures have a *p*-fold rotational symmetry and have \((p-2)\) different perimetral densities. On the other hand, Bracho, Montejano, and Oliveros [173] proved that if the perimetral density \(\sigma \) is \(\frac{1}{3}\) or \(\frac{1}{4}\), then the solution is circular. Later, in [1182], F. Wegner was able to give nontrivial explicit solutions to this 2-dimensional version of the floating body problem.