Aside from basic books and surveys (see [160], [1204], [238], and [527]), there are also various expository papers summarizing and nicely presenting geometric properties and characterizations of constant width figures; an example is [598]. We now want to collect further geometric properties and characterizations of figures of constant width in the plane, which are widespread in the literature.

**Characterizations and Properties**

In view of the importance of binormals (see Chapter 3), we start these notes with related characterizations of figures of constant width. As starting point, we should mention the papers [510], [511], [512], [501], [507], [508], and [509], where a thorough study of diametrical chords of constant width figures (also for Minkowski planes) is given. Theorem 3.1.6 (Theorem 4.1.1) goes back to Bückner [194], and based on this he observed also that a 3-dimensional convex body *K* has constant width if and only if the line perpendicular to every pair of nonparallel normals of *K* intersects *K*, see also Exercise 3.10. Extending another result of Bückner [194], Beretta and Maxia [99] proved that a strictly convex figure *K* is of constant width if and only if for all directions *u* the expression \(2f(u)-w(u)l(u)\) is constant; here *l*(*u*) and *f*(*u*) denote the length of the boundary as well as the area of a part of *K* on the same side of a diametrical chord of direction *u*, and *w*(*u*) is clearly the respective width of *K*, see Corollary 5.1.5. In [866], it is shown that a convex planar set is of constant width if and only if its boundary is the image of a continuous mapping of a circle, such that diameters are mapped to double normals. Kelly [617] proved the following: If *C* is a constant width figure in the plane, its pedal curve \(C'=p(C)\) with respect to an arbitrary point *a* has *a* as an equichordal point. Conversely, given a curve star-shaped with respect to its equichordal point *a*, its “negative pedal curve” \(C = p^{-1}(C')\) (easily obtained as an envelope of straight lines) has constant width if it is convex. The author studies non-convex curves obtained in this way which are inner parallel curves of constant width curves. In [778], the results from [617] are extended to higher dimensions. For example, it is shown that a smooth convex hypersurface with the origin as an equichordal point has an antipedal hypersurface which is a hedgehog of constant width (for hedgehogs we refer to the notes in Chapter 11). A condition is given ensuring that the pedal hypersurface of a given hedgehog of constant width is convex.

Theorem 4.1.3 can also be formulated as follows (see [531]): A closed convex curve is of constant width if and only if each of its chords is the longest chord of one of the two arcs determined by its two endpoints. Inspired by the fact that each planar convex body has an inscribed square, Eggleston [314] constructed examples which limit generalizations of this result. He showed that there is a constant width figure in which no regular *k*-gon may be inscribed for \(k > 4\) (in fact, the Reauleaux triangle is shown to have this property). The author of [250] derived an exact condition for the existence of a regular polygon circumscribing a strictly convex figure in the plane. His result yields a generalization of Pál’s old theorem (a plane figure of constant width has a circumscribed regular hexagon) in the following way: If the width function of a strictly convex figure *K* has period \(\pi /3\), then there is a regular hexagon circumscribing *K*. In [1133], constant width curves are characterized by the property that, if three edges of a rhomb belong to lines of support of the curve, then also the fourth. If “rhomb” is replaced by a quadrilateral which is the union of two isosceles triangles, the condition characterizes circles. Let *K* be a convex body, and *F* be a set of pair-wise intersecting translates of *K*. It is known that if *K* is centrally symmetric, then for any *F* one can find three points which meet every member of *F*. The conjecture that this is also true if *K* is not centrally symmetric is proved in [242] for *K* being a constant width figure. This result follows from the next interesting theorem: If *Q* is a set of diameter 1 and *K* a figure of constant width not less than 0.9101, then *Q* is contained in the union of three translates of *K*. In [513], it is shown that a closed curve of length \(2\pi \) can be covered by a rectangle of area 4, and if no smaller rectangle will do this, the covered curve has to be convex and of constant width 2 (see also [304]). More analytic characterizations of constant width curves are given in [409], [935], and [866].

In [878], planar curves are studied from which any point *p* has another point \(p'\) of constant distance, such that the oriented tangents at *p* and \(p'\) have opposite directions (and the spherical image of the curve should be closed). Such curves are shown to be either of constant width or to be so-called translation curves consisting of a set of congruent arcs. In [607], closed convex curves (and, in particular, polygons) are investigated all whose circumscribed rectangles are squares. Clearly, constant width curves occur as examples. Let *C* be a body of constant width *h* in Euclidean *n*-space. In [377], it is proved that the lengths of the segments intercepted by *C* on each of two parallel lines at most *v* distance apart differ by at most \(2(2vh)^{1/2}\), where this value depends on the width, but not on the shape of *C*. And in [621] the following is investigated: A convex body *K* in \({\mathbb E}^n\) has property (P), if for every direction *u* and every pair of parallel supporting hyperplanes parallel to *u* there is a chord of maximal length in the direction of *u* lying midway between the hyperplanes. Is any *K* satisfying (P) symmetric? (The converse is obvious.) An affirmative answer is given for general convex bodies if \(n \ge 3\), and for constant width figures if \(n = 2\). Thus, for general convex bodies, the question is unsettled only in the planar case.

There are also results characterizing special curves (e.g., disks) within the family of constant width figures, or describing at least respective properties. One old result in this direction is obtained in [511]: A constant width figure, whose boundary is divided by each diametrical chord into equal arcs is necessarily a circular disk. Assuming smoothness, this was proved already by Hirakawa [541] and independently by Klamkin (see again [511] and also [104]). On the other hand, it was also proved in [511] (and also with smoothness assumption already in [541]) that if any diametrical chord of a constant width figure bisects its area, then it is again a circular disk. See Corollary 5.3.1 for both results. For more on these two characterization theorems and related circle characterizations, we also refer to [829] and to [407], where it is proved that the only Zindler curve of constant width is the circle, see Exercise 5.19. Based on *X*-rays, in [724] a sufficient condition for a constant width figure is given to be a disk. The author of [162] proved that, if *C* is a constant width curve, such that for each pair *x*, *y* of points from *C* there exists a nondegenerate rectangle *R* contained in *C* with *x*, *y* being also from *R*, then *C* must be a circle. The paper [777] contains results on hedgehogs, applications which yield characterizations of spheres among all convex hedgehogs of constant width. In [967], the following problem is raised: Put \(f(x, y)=1/\pi (1-x^2-y^2)^{1/2}\) for \(x^2+y^2 < 1\) and \(f(x, y)=0\) for \(x^2+y^2 \ge 1\). Clearly, the integral of *f*(*x*, *y*) on every chord of the unit circle is 1. Is there any other curve of constant width for which there exists a function *f*(*x*, *y*), whose integral is constant on every chord? Baillif [63] derived a characterization of the circle among all \(C^2\) curves of constant width via inscribed regular polygons.

**Mizel’s Conjecture**

The following notes refer to Section 4.2. In [1213], Zamfirescu proved an analogue of Theorem 4.2.1 for a Jordan curve (not convex a priori) and for a rectangle with the infinitesimal rectangular property satisfied by the following relation between its sides: \(|a/b| \le \varepsilon , \varepsilon > 0\), where *a* and *b* are the side lengths of the used rectangle. The usual rectangular property is given by \(\varepsilon = 1\). In 2006, M. Tkachuk extended this infinitesimal rectangular property to arbitrary compact planar sets, whose complements are not connected. Zamfirescu also showed that every analytic curve of constant width satisfying the infinitesimal rectangle property is a circle. The main result of the paper [1217], where Tkachuk is one of the authors, is to prove this theorem for a convex curve which is not analytic, and to discuss some further unsolved problems concerning Mizel’s problem. Wegner [1177] considered extensions to plane continua and higher dimensional manifolds. He showed that any planar continuum having the above rectangle property and which is not a Jordan arc must be a circle. Further on, any bounded subset of the plane with the rectangle property and containing a circle must in fact be a circle. For higher dimensions it is noted that there exist smooth closed curves in 3-space having the rectangle property, which is not circles. On the other hand, it is proved that a compact hypersurface immersed in *n*-space and having the rectangle property must be a sphere. Further extensions to curves and hypersurfaces in higher dimensions are given in [1178]. For example, it is proved that a \(C^2\) simple closed curve in 3-space satisfying the infinitesimal rectangular property must be a transnormal curve (as generalization of the notion of constant width curves, see Section 16.2). If the curve satisfies the rectangle property then it is a transnormal, centrally symmetric, spherical curve. On the other hand, any \(C^1\) transnormal, centrally symmetric, spherical curve satisfies the infinitesimal rectangular property. Several other related results, also referring to higher dimensions, are proved in [1178], assuming also weakened forms of the infinitesimal rectangular property.

**Alexander’s Conjecture**

Ault [49] verified Herda’s conjecture (see Section 4.3) extended to any real Hilbert space by using simply the law of cosines for a triad of triangles, a few elementary trigonometric formulae, and a generalized triangle inequality, thus making the proof rather easily understandable. Studying closed curves and actions of finite groups on differential-geometric manifolds in *n*-space, Aeppli [3] re-obtains results such as Theorem 4.3.2. His approach also allows to obtain a related characterization of spheres in 3-space. Lutwak [741] succeeded to use Falconer’s result from [329] for showing the following: Any closed curve of length 1 can be covered by a semicircular disk of radius \(1/\pi \). Moreover, this radius cannot be decreased if and only if the curve is of constant width \(1/\pi \). From this, it follows that any two planar domains, each bounded by a simple closed curve of length 1, may be placed inside a circle of radius \(1/\pi \), such that their interiors are disjoint. Two figures of constant width \(1/\pi \) show that the radius of the circle cannot be decreased, and it is proved in [741] that there are no other cases of this kind.