Many results in convexity refer to convex bodies with certain smoothness assumptions. Then, as it is natural, the used methods are often taken from differential geometry. This is done also in this chapter here, but other parts of chapters (e.g., Section 16.2) refer to that direction as well. Basic survey-like references closely related to this part of convexity are [160, Chapter 8, Chapter 17], [712], and [713]. In particular, curves and surfaces of constant width are also interesting from the viewpoint of differential geometry. Books from that field taking special care for constant width sets are [1147, Chapter I], [830, Chapter 3], [561, Chapter 2], and [18, Chapter 5] (see also [1104]). Regarding classical curve theory we also mention the impressive book [731, pp. 311–316]. This chapter here, together with its notes, is strongly related to our Chapter 5 about line families (topics like pedal curves, evolutes/involutes, envelopes, hedgehogs etc. are discussed there) and, via curvature notions, to the Chapter 12 on mixed volumes.

**Curves of Constant Width**

We start these notes with some *elementary theory of curves* (of constant width), later on coming also to higher dimensions and special topics (mainly that of curvatures and that of spatial curves of constant width).

Braude [184] investigated cycloids of constant width curves, or used them as cycloids to generate new curves (including the case that a constant width curve rolls on a constant width curve). Santaló [1013] showed how to generate curves by an endpoint of a segment which moves (with the other endpoint) on a closed, arc-length parametrized curve. It turns out that a special case of this process yields curves of constant width. In [556], a cam mechanism with two rigidly connected disks is studied, leading to curves of constant distance sum of corresponding tangents. These can be interpreted as generalized constant width curves, and a kinematic generation additionally yields generalized Zindler curves, see Section 5.4. On these lines, the author obtains also a generalization of Barbier’s theorem. In [995], pairs of closed Bertrand curves, Frenet formulas, and an inequality of Fenchel are used to derive Barbier’s theorem. In [955] envelopes of lines and the global properties of some special one-parameter equiform motions are discussed, yielding also generalizations of Holditch’s and Barbier’s theorem. Using dual line coordinates, in [11] Bertrand offsets of ruled surfaces in view of their dual representation are examined. As a limit position of closed ruled surfaces, also results on closed space curves are derived. In this framework, Barbier’s theorem is reproved, and the results are illustrated by computer-aided examples. The paper [769] presents a deep study of Zindler curves, using constructively also their close relations to constant width curves; e.g., rotating the double normals (meant as chords) of a constant width curve about \(\frac{\pi }{2}\) and around their midpoints, one gets the main chords of a Zindler curve (see also [1102] and Section 5.4.3). This is also related to our notes on transnormality in Section 16.2. However, Wunderlich developed in [1201] a kinematic principle which is based on the family of tangent lines of the midpoint curve, allowing a spatial construction of Zindler curves without using spatial curves of constant width (and hence not done as in [555]). In [1125], osculating circles of smooth constant width curves (having the same width as these circles) are considered. It is shown that there exist at least three such osculating circles which cross the constant width curve exactly twice, both times tangentially, and that these circles are the usual osculating circles at each of the crossing points. Mozgawa [870] generalized Mellish’s theorem (see below under “curvatures”) about constant width curves in the following way: He introduced curves of constant \(\alpha \)-width, defined with the help of a curve called the *isoptic *\(\alpha \)*-curve* associated to the given curve *C*, with \(\alpha \in (0,\pi )\). This isoptic \(\alpha \)-curve is the curve consisting of points at which two tangents to *C* meet at an angle of \(\pi - \alpha \). The \(\alpha \)-width of *C* is defined as the curvature of this associated isoptic \(\alpha \)-curve. Clearly, in the limit (when \(\alpha = \pi \)) the \(\alpha \)-width coincides with the usual width. In fact, the concept of isoptics is much older: Nitsche [896] called a planar curve *C* an *isoptic for a circle K* if *C* subtends the same angle at all points of *K*, and he proved that *C* is a circle if it occurs as isoptic for two distinct concentric circles. In [257], results on rosettes (meaning plane \(C^2\) curves with everywhere positive curvature) are derived, where certain pairs of points (such as antipodal points, orthodiameter points,...) in such a curve are introduced and studied in their relation to each other. It is shown that rosettes of constant width (as generalizations of constant width curves) have necessarily odd winding numbers, and that rosettes of constant width *h* and winding number *k* have perimeter \(\pi h k\). More general results in this direction are presented in [1174], where the author also extended classical constructions of convex curves to rosettes of constant width. More general results on rosettes were obtained in [262]. In [828], the notion of isoptics is adjusted to that of rosettes, yielding again rosettes of constant width. Their length is computed, it is shown that the number of axes of symmetries for rosettes of constant width has to be odd, and that central symmetry is only possible for circles. Also in [255], [256], and [828] isoptics of plane closed convex curves are investigated, yielding again results about constant width curves in this framework. Isoptics as a topic are more than hundred years old; their generation is also of interest in the theory of mechanisms. For example, in [256] the preservation of constant width is investigated. And also in [828] various basic theorems on constant width curves are extended to rosettes of constant width. Porcu [947] proved several metric and affine properties of so-called *P*-curves (these convex curves, going back to Blaschke, have a simple infinity of inscribed quadrangles of maximal area) and curves of constant width and their affine and projective transforms. In [259], a six vertex theorem for equipower curves (as certain “counterpart” of constant width curves) is proved. Related also to [231], the authors of [871] study some geometric problems concerned with plane sets of constant width relative to a certain given oval. In particular, they prove a six vertex theorem for such curves. Also the paper [899] is concerned with vertex theorems of plane curves, yielding results on the perimeter of equiangular polygons circumscribed about constant width curves. Further results on the differential geometry of plane curves of constant width were derived in [735] and [995].

Motivated by CAD software applications, the authors of [8] used piecewise Pythagorean hodograph curves to construct curves and rosettes of constant width; the concept of involutes plays a role there. Related is the paper [7], in which the well-known concept of offset curves is extended to normed planes and where again (analogues of) Pythagorean hodograph curves are used, since their offset curves are rational and allow flexible interactive design. In this way the authors carry over tools and notions like Serret–Frenet equations, evolutes, and involutes to normed planes. Related, since also referring to constant width in normed planes, are the papers [269] and [270], where also the concepts of hedgehogs and involutes occur.

Using smoothness of support functions, Kharazishvili [620] showed the following approximation result: if a closed convex curve of constant width has everywhere strictly positive curvature and a support function belonging to the class \(C^3([0, 2 \pi ])\), then, for any \(\varepsilon > 0\), the \(\varepsilon \)-neighborhood of this curve contains an algebraic curve of constant width. The paper [393] deals with an evolution procedure of strictly convex plane curves taking their tangential angle, width, and curvature radius suitably into consideration. After showing area-increasing and perimeter-preserving properties of this flow (under which constant width curves remain invariant), it is proved that under certain assumptions the limit curves are of constant width or (if the initial curve is centrally symmetric) even circles.

Now we switch to the concept of *space curves of constant width*, see also Section 16.2. There are several definitions, also depending on the dimension. We will start with the following classical one which is perhaps also most common. A \(C^1\) curve embedded in \({\mathbb E}^3\) is a space curve of constant width *h* if every normal plane of this curve intersects it in exactly two points which are antipodal and of distance *h* apart. Later on we will discuss also space curves in higher dimensions. It seems that Fujiwara started the topic with the paper [378], giving also the definition above. Answering a question of him, Blaschke [129] showed that such a curve has to lie in the boundary of a constant width body, see also [160, p. 139]. This was reproved by Bückner [192], who also showed that any space curve of constant width *h* has length at least \(\pi h\) (see [194]). General classes of space curves in \({\mathbb E}^3\) (with those of constant width as special cases) were investigated by [1004] and [878].

It is well known that the length of a space curve of constant width *h* in \({\mathbb E}^3\) satisfies \(l(C) \ge \pi h\). Using a result from [1167] (asserting that any such curve can be isotopically deformed through curves of constant width *h* to a curve on a sphere of radius *h*) and Crofton’s formula for spheres, the author of [1123] derived the non-sharp upper bound \(l(C) < (3 \sqrt{2})\pi h\). In [1167], the above mentioned deformation is shown to preserve length, integral curvature and integral torsion, whereas the variance of the geodesic curvature is minimized by the spherical curve. It is also proved that any nonplanar curve of constant width has at least 12 geodesic vertices. Also on these lines, Wegner [1169] proved that a spherical curve of constant width *h* lies between two parallel planes having distance \(\frac{h}{\sqrt{3}}\), and that a curve of constant width is isotopic to the circle through curves of constant width. In [247], the author established some nice formulae relating the perimeter, curvature, and torsion of space curves of constant width in 3-space to their width. In [1004], several characterizations of such curves are obtained, via Gauss curvature, point-tangent distances, angles between “opposite tangents”, and further properties. Pottmann [951] investigated spherical motions associated to spherical curves in \({\mathbb E}^3\) and also in higher dimensions, and he found, within this framework, also applications to spherical curves of constant width. Further results about curves of constant width in \({\mathbb E}^3\) were obtained by Nádeník [881]. Armstrong [37] investigated certain types of space curves (e.g., geodesics, contact curves with circumscribed cylinders, lines of curvatures, planar sections, etc.) in surfaces of constant width in \({\mathbb E}^3\) and showed, which pairings of such properties imply *all* properties satisfied by these curves. Sezer [1057] characterized space curves of constant width via differential equations, and in [653] it is shown that the integral torsion of a space curve of constant width is an integer multiple of \(2 \pi \).

Staying with space curves of constant width, we go now to higher dimensions. After a historical survey, Nádeník studied in [882] space curves of constant width in even-dimensional Euclidean spaces, with constant ratios of curvatures. He then proved several theorems on such curves, e.g., on their lengths, relations between opposite points, centers of gravity, and particular types of such curves. In [142], pairs of closed curves in \({\mathbb E}^n\) having the same spherical images (obtained via normals) are studied. Using also the arc length of the spherical image, theorems on the difference or sum of their perimeters were obtained, generalizing known properties of constant width curves. Nádeník [883] presents eleven properties of space curves in \({\mathbb E}^4\), again determined via such spherical images and guaranteeing constant width. Also in [645] and [749] constant width curves in 4-space are studied. S̆makal [1071] derived necessary and sufficient conditions for space constant width curves in \({\mathbb E}^n, n > 3,\) with symmetric spherical images of their tangent vectors, yielding also a natural analogue of Barbier’s theorem and of Meissner’s theorem concerning the coincidence of the centroid and the Steiner center of curvature.

**The Steiner Point and Related Topics**

The center of mass of a uniform mass distribution on a convex curve \(\mathcal {C}\) is the *perimetral centroid*, and the center of mass of a distribution on \(\mathcal {C},\) whose density at each point is equal to the curvature, is called the *Steiner point*; see Section 14.5, where the Steiner incircle will be used to generalize the isoperimetric inequality in the plane. It is known that if \(\mathcal {C}\) has constant width, the perimetral centroid and the Steiner point coincide [882] (see also [169]; we refer to [882] for an analogue referring to space curves of constant width). Furthermore, the locus of the perimetral centroid of the outer parallel curves of a curve of constant width lies on a straight line. In [273] the areas of pedal curves with respect to Steiner points are used to get an inequality for constant width sets. Ganapathi [390] studied those curves whose outer parallel curves all have the same curvature axis (defined via mass distribution whose density is the curvature); he showed that all of them have at least six vertices, thus reproving the fact that each constant width curve has at last six vertices, see Theorem 11.3.3. Soloviev [1073] proved that the support function of any convex curve whose centroid coincides with its curvature centroid is representable by \(h(u) = s(u) + cg(u)\), where *c* is a constant, *s*(*u*) is the support function of a centrally symmetric curve, and *g*(*u*) is the support function of a parallel curve of a constant width curve. The following was shown in [1132]: A closed, connected, orientable surface *W* in \({\mathbb E}^3\) is called a surface of generalized constant width *h* if it satisfies the following conditions: (1) let *n*(*p*) be its inward unit normal at \(p \in W\); then the vector \(p' = p + h n(p)\) lies on *W*; (2) the map \(p \rightarrow p'\) is an involution. It is proved that if *W* is an analytic surface of generalized constant width *h* and the Gauss curvature of *W* at *p* equals that of \(p'\) for every \(p\in W\), then *W* is a sphere of diameter *h*. The author of [1145] studied differentiable hypersurfaces which have exactly two normals parallel to every direction. He confirmed properties known for bodies of constant width, but gave also more general statements. In [1035], Schneider generalized most of the classical real-valued functions associated to convex bodies to vector-valued functions. He proved that hereby many well-known relations either carry over exactly or slightly modified. As from the notion of volume in a natural way the notions of mixed volumes and quermassintegrals are obtainable, one can derive from the centroid a series of curvature centroids, among them the surface area centroid and the Steiner point. In the final section of [1035] it is shown that a whole class of linearly independent relations involving one of the vector-valued functions and the quermassintegrals holds for sets of constant width. For the spherical situation, similar results were derived in [38], and analogues for envelopes of cylindrical surfaces of constant width are given in [884].

**Curvatures**

We now come to the important notion of *curvature*. Beretta and Maxia [99] showed that for curves of constant width the number of vertices (i.e., of points of extreme curvatures) is of the form \(4n+2, n>0\), when the radius of curvature is assumed to be continuous. The obvious fact of nowhere vanishing curvature was proved in [261]. Mellish [814] and Hsiao [560] proved several properties of principal directions and radii of curvature as well as mean curvatures at opposite points (defined via parallel normals) of curves and surfaces of constant width, see Theorem 11.3.1. Here we refer also to [261] and [66].

One can generalize Theorem

11.3.1, 3) to all convex bodies without smoothness restrictions by introducing the first curvature measure

\(S_1(\Phi , E)\). See Section

12.2. For a body

\(\Phi \) of constant width

*h* we have that

\(\Phi +(-\Phi )=hB\). Therefore, by linearity and homogeneity of

\(S_1(\Phi , E)\) as a function of

\(\Phi \), we obtain

$$\begin{aligned} S_1(\Phi , E)+S_1(-\Phi , E)=h \gamma (E), \end{aligned}$$

(11.9)

where

\(\gamma (E)\) is the spherical Lebesgue measure of the Borel set

\(E\subset \mathbb {S}^{n-1}\). Thus, if

\(\phi \) is a convex body satisfying (

11.9) for all Borel sets

\(E\subset \mathbb {S}^{n-1}\), then

\(S_1(\phi +(-\phi ), E)=S_1(hB, E)\), and by the Aleksandrov–Fenchel–Jessen Uniqueness Theorem (see Schneider [1036]) we obtain that

\(\phi \) has constant width

*h*. Thus (

11.9) is a characteristic property of bodies of constant width

*h*.

For a very interesting treatment of the curvature of convex surfaces we may consult a paper by Mellish [814]. This paper, although it appeared in the Annals of Math., is not well known, maybe because the words “constant width” do not appear in the title. Mellish worked on a generalization of Barbier’s Theorem. He died at the age of 24 and had no mathematical publications during his lifetime. After his death, his colleagues at Brown University examined his notes on mathematics and prepared this paper based upon his notes. Formula (11.5), which clearly resembles Euler’s Sectional Curvature Theorem 11.2.2, was first proved by Blaschke in his book “Kreis und Kugel” (see [132], pp. 117). Mellish’s proof is different from that of Blaschke and was obtained independently. With respect to the curvature of a surface of constant width, Hsiao proved in [560] an interesting theorem involving the sectional curvature and the Gaussian curvature at the endpoints of a diameter (see Theorem 11.3.1 4).

A general *n*-dimensional convex body shows a regular curvature behavior at almost all its boundary points. Due to a theorem by Aleksandrov [13] almost all boundary points of a convex body are, in the sense of measure, normal points. Hence all sectional curvatures exist, and they satisfy Euler’s Theorem. This is true even without differentiability assumptions. From the generic point of view the picture is as follows: while a typical convex body is, in the Baire category sense, strictly convex and smooth, it was proven by Zamfirescu [1214] that at almost all its boundary points the curvature is zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. In analogy to this result it is to be expected that for a typical convex body of constant width 1 the radii of curvature exhibit a preference for attaining the values 0 and 1. Indeed, it was also shown by Zamfirscu [1215] that for a typical convex figure of constant width 1 in the plane the radii of curvature attain only the values 0 and 1, see Theorem 11.4.3. A higher dimensional extension of this result was given by Bárány and Schneider [73]; they proved that the typical behavior of convex bodies of constant width 1 is that for almost all boundary points all curvatures are equal to 1. The proof of this theorem requires a more elaborate approximation procedure than Zamfirescu’s result. Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.

**The Local Geometry of a 3-Dimensional Convex Body**

For the local geometry of the boundary of a 3-dimensional convex body (see Section 11.5.2) two papers are essential. The first is due to Bayern, Lachand–Robert and Oudet [86], and the second due to Anciaux and Guillfoyle [31]; they apply methods of variational analysis to study the Blaschke–Lebesgue Conjecture.

If

\(\phi \subset \mathbb {E}^2\) is a convex figure which has sufficiently smooth boundary,

\(\rho (\theta )\) denotes the radius of curvature of

\({{\,\mathrm{\mathrm {bd}}\,}}\phi \) at the point with outward normal unit vector

\(\mathbf {u}(\theta )\) and, in addition,

\(\phi \) has constant width

*h*, then (

11.9) becomes

$$\begin{aligned} \rho (\theta )+\rho (\theta +\pi )=h\,, \end{aligned}$$

see Theorem

5.3.7. This equation is only a necessary condition on a given function

\(\rho (\theta )\) if we want to characterize the radius of curvature function of the boundary of a constant width function. Kallay [603] assigns a radius of curvature function to every figure

\(\Phi \) of constant width

*h* and gives necessary and sufficient conditions for a nonnegative and measurable function

\(\rho (\theta )\) to be the radius of curvature of some figure of constant width. He uses this to characterize the “extreme bodies” among figures of constant width that are indecomposable with respect to Minkowski addition. Kallay proved that a convex figure is extreme in this sense if and only if the radius of curvature takes the values 0 and 1 for almost all

\(\theta \). For example, Releaux polygons (see Section

8.1) of width 1 are among the extreme bodies. A counterpart of Theorem

5.3.6 in Minkowski planes is given by Petty [929]. Vicensini [1152] gave an extensive treatment of related matters, including the analogue of Theorem

11.3.1 3) in Minkowski spaces.

**Affine Geometry**

Convex bodies of constant affine width were also studied from the viewpoint of affine differential geometry. This area was started by Süss [1108] in 1927. An excellent survey of the basic results in case \(n=2\) can by found in Beretta–Maxia [99], including the discussion of Hirakawa’s work [542] about the concept of affine width. Furthermore, for more related material on the affine geometry of convex curves see Heil [521]. An interesting affinely invariant concept was introduced in [1006]: For a plane convex curve *C* and each \(p \in C\) let *t*(*p*) be the maximal area of all triangles inscribed in *C* having *p* as vertex. Then *t*(*p*) is the triangular width of *C* at *p*, and *C* is said to be of constant triangular width if *t*(*p*) is constant for all \(p \in C\). In [1006] and [1007], such curves (different to ellipses) are constructed, and higher dimensional analogues are given. In [1008], a characterization of ellipses inside that class of planar curves is presented. Further on, in [1009] it is clarified which regular polygons can have this property. Furthermore, in [263] and [395] it was shown that 4-gons and 5-gons of constant triangular width must be affine images of regular polygons, but 6-gons and 7-gons not. And in [394] these considerations are continued, where some parallelism between sides and diagonals of polygons of constant triangular width is established as tool for possible further research on such notions. Using affine diameters of convex sets, Alonso and Spirova [25] introduced a new orthogonality type for normed planes, called affine orthogonality. This concept leads to characteristic properties of sets of constant width (with respect to the considered norm).

Leichtweiss [710] considered inequalities referring to the ratio of the radii of smallest circumscribed and the largest inscribed spheres of *n*-dimensional convex bodies ranging over the class of all their images regarding affine transformations. If such a class contains a constant width set, the infimum case yields a characterization of constant width sets.

An *n*-dimensional convex body is said to have *constant diagonal* if the main diagonals of all its circumscribed boxes (i.e., rectangular parallelotopes) have constant length. Chakerian [229] proved that for \(n > 2\) this property characterizes all affine images of constant width bodies. And we have also the affine image of a constant width body iff all its orthogonal projections onto hyperplanes are affine images of a body of constant width within this hyperplane, see Section 11.6. For the planar case, the second condition is empty, and the first one is only necessary. Krautwald [655] obtained nice characterizations of affine images of *n*-dimensional constant width bodies. One of them says the following for a convex body *K*: let *P* be combinatorially equivalent to a cross-polytope and inscribed in *K*, having maximal volume among all such polytopes. Assume that all its vertices sit in the interiors of the facets of a suitable parallelotope circumscribed about *K* and having minimal volume among all parallelotopes circumscribed about *K*. Then *K* is the affine image of some constant width body. The paper [947] contains several results about generalized Radon curves and affine as well as projective images of (planar) constant width bodies. From results of Blaschke on ellipsoids it follows that the affine images of 3-dimensional constant width bodies are characterized by the property of having projections bounded by *P*-curves. See Exercises 11.17 and 11.18. Applying the Blaschke–Santaló inequality to the difference body of arbitrary convex bodies, Lutwak [742] derived an inequality involving *n*-means, diameters, and widths of them in a direction *u*, where equality holds for affine images of constant width bodies. Inspired by the coat of arms in the castle of Blois consisting of a packing of Reuleaux triangles, in [343] convex sets of direction-invariant packing density in the plane are investigated. Affine images of squares and constant width sets belong to this family. Let *L* denote a system of *m* lines in the plane. Depending on *m*, Alexander [19] studied homothety ratios for affine images of constant width sets in the plane such that their smaller homothets are not met by any line from *L*, whereas each line of *L* meets the original set. In [1005], pairs of affine equidistant curves in 3-space are introduced; when these pairs of curves coincide, spatial curves of constant width are obtained.

Another related notion concerns the polar of a body of constant width. Let

*o* be an interior point of the convex body

\(\Psi \subset \mathbb {E}^n\). We say that

*o* is an

*equichordal point * if all chords of

\(\Psi \) passing through

*o* have the same length. The point

*o* is a

*equireciprocal point* if, for every chord

*pq* of

\(\Phi \) through

*o*, the quantity

$$\begin{aligned} \frac{1}{|op|}+\frac{1}{|oq|} \end{aligned}$$

is constant. Both concepts are wider discussed in the Notes of Chapter

13 below (see the part below the sub-headline “Constant section”).

It is possible to prove that if a body of constant width \(\Phi \subset \mathbb {E}^n\) has the origin *o* as an interior point, then its polar body \(\Phi ^o\) has the origin *o* as an *equireciprocal point. * Furthermore, if *S* is the surface obtained from \({{\,\mathrm{\mathrm {bd}}\,}}\Psi \) by inversion with respect \(\mathbb {S}^{n-1}\), then the origin *o* is a equichordal point of *S*.

The above theorem is of interest with respect to the famous problem of Fujiwara asking whether there exists a plane convex figure with more than one equichordal point. Despite its elementary formulation, this problem remained unsolved for many years until it was finally proven in 1996 by Rychlik [993] who showed that such a figure with two equichordal points cannot exist. He used methods of advanced complex analysis and algebraic geometry. A discussion of this problem and related problems about equichordal points is given by Klee, see [631], [634] and also the notes to our Chapter 13.