**Helly’s Theorem and Discrete Geometry**

A prominent role in combinatorial geometry is played by Helly’s theorem, it has stimulated numerous generalizations and variants. There are many interesting connections between Helly’s theorem and its relatives, the theorems of Radon, of Carathéodory and of Tverberg as we partially mentioned already in Section 2.8. These theorems have been the object of active research, and they inspired many problems in the field of discrete geometry. We mention here the excellent surveys [278] and [306] to see a sample of numerous problems associated to Helly’s theorems.

One of the most beautiful theorems in combinatorial convexity is due to Tverberg, that is, the *r*-partite version of Radon’s Lemma. To be more precise, Tverberg’s theorem states that every \((n+1)(r-1)+1\) points in Euclidean *n*-space \(\mathbb {E}^n\) can be partitioned into *r* parts such that the convex hulls of these parts have nonempty intersection, see [1138]. This theorem still remains central and is one of the most intriguing results of discrete geometry. It has been shown that there are many close relations between Tverberg’s theorem and several important results in mathematics, such as Rado’s Central Theorem on general measures, the Ham-Sandwich Theorem, and the Four-Color Theorem, just to mention some examples.

Tverberg’s theorem is closely connected with the multiplied or colorful versions of the Theorems of Helly, Hadwiger, and Carathéodory, first studied by Bárány and Lovasz, see [40]. In fact, there is a topological version of Tverberg’s theorem that has received much attention in the last decades. During the 90s, techniques and ideas of algebraic topology were used in a relevant and deep manner to study this version, and nowadays, due to the influence of Gromov’s topological ideas, the late developments of this problem became an important area of research. See [74] and the beautiful book of Matoušek [802].

Helly-type theorems explicitly referring to constant width sets can be found in [29] and [30]. For example, in [30] it is proved that if every \(n+1\) sets of a family of compact convex sets in *n*-space simultaneously contain a set of width \(h > 0\) (respectively, a set of constant width *h*), then all members of this collection contain a set of constant width \(h_1\), where \(h_1 = h/\sqrt{n}\) if *n* is odd, and \(h_1 = h \sqrt{n+2}/(n+1)\) if *n* is even. The author calls such a set a common set (of constant width \(h_1\)) of the whole family.

**Universal Covers**

The notion of *universal covers* clearly plays an essential role regarding Borsuk’s partition problem (namely, by dissecting universal covers of diameter-one sets into pieces each having diameter less than 1); see [488], [272, D 15], [151, Chapters V and VII], and [182, Section 11.4]. A nice survey on constant width sets mainly devoted to universal covers (but discussing also other problems from discrete geometry) is [1192], also updating [238]. To construct universal covers having constant width sets in mind, Hansen [514] used coverings of Reuleaux polygons by polygons. Chakerian and Logothetti [239] proved that the smallest convex *m*-gon (\(m > 3\)) being a universal cover coincides with the smallest regular *m*-gon that covers the Reuleaux triangle of width 1. This is based on Pál’s classical result that a plane constant width set has a circumscribed regular hexagon [909] (see Theorem 15.2.3). For solving a problem of Klee, Eggleston [315] showed that the known union of a circular disk and of a Reuleaux triangle, both of unit diameter and placed so that two of the vertices of the Reulaux triangle are contained in the disk, is a universal cover, see Section 15.2.4. Makeev [758] proved the following: let *M* be the intersection of finitely many half-spaces in \({\mathbb E}^n\), and *X*(*M*) be the union of all bodies of constant width 1 contained in *M*. Then the polyhedral set *X*(*M*) is a semi-algebraic set, and by this construction a possible universal cover *M* can be made smaller by substituting *X*(*M*) for *M*. Also the paper [757] refers to universal covers and constant width bodies. Namely, a value \(m(\varepsilon )\) is established such that, for \(m > m(\varepsilon )\), each constant width body in \({\mathbb E}^n\) has an \(\varepsilon \)-aspherical orthogonal projection of dimension *m* (meaning the deviation from the spherical shape measured via the ratio of diameters of in- and circumspheres). The proof is based on universal covers considered in \({\mathbb E}^m\) which are *m*-polytopes with 2*m* facets. Related is also the exposition [760], presenting results on 3-dimensional polytopes having homothets that are inscribed in or circumscribed about a 3-dimensional compact set. The cases with 3, 4, 5, 6, and 7 facets are discussed in detail, and also the problem of polyhedra circumscribed about constant width bodies is taken care of. Related results are also presented in [762]; e.g., for \(n = 3\) two one-parameter families of polytopes with 12 facets circumscribed about any constant width body are presented.

**Packing and Covering**

We start with *packings*. As we already know, for an open convex subset *G* of \({\mathbb E}^n\) and \(p \in {\mathbb E}^n\), \(G \cap (G+p) \ne \emptyset \Leftrightarrow G^* \cap (G^* + p) \ne \emptyset \) holds (see [238, p. 77]), where \(G^*\) denotes the central symmetral \(\frac{1}{2}(G + (-G))\) of *G*. This equivalence allows the reduction of many translative packing problems for constant width bodies to corresponding problems for balls (taking *G* as their interior), and thus literature about ball packings is also relevant here (see, for example, the excellent surveys [341], [342], [338], and [339], as well as the book [163]). To give an example: finding the densest translative packings of constant width bodies in \({\mathbb E}^n\) is equivalent to finding the densest corresponding ball packings. (Unfortunately, there is no corresponding reduction for analogous covering problems; see [453].)

For a convex body *K* in \({\mathbb E}^2\), let \(K_1, K_2\) be two nonoverlapping smaller homothets of *K* contained in *K*. It is easy to see that if *K* is of constant width, then for the perimeters the inequality \(p(K_1) + p(K_2) \le p(K)\) holds. Somehow conversely, Beck and Bleicher [94] showed that the following holds: if for \(K, K_1, K_2\) being such convex bodies in the plane the same inequality holds, then *K* is either of constant width or a regular polygon. They also considered the concept of constant minimal width, see Section 3.2.

Finite packings of Reuleaux triangles are sometimes nicely visible in Gothic church windows, see Figures 1.4 and 18.2. The authors of [138] give an estimate for the packing density of the Reuleaux triangle, understanding it as a good example for investigating packings of nonsymmetric figures. To find the densest translative packings of Reuleaux triangles is not hard, but if congruent copies are allowed, the problem is difficult and (to our best knowledge) unsolved. Continuing [138], in [139] the case is studied when the Reuleaux triangles are not necessarily translates of each other, but the packings still satisfy a certain regularity condition. If *I* is one of the 2-dimensional discrete groups of isometries and no two copies of a Reuleaux triangle *R* following *I* overlap, then we have an *I*-packing of Reuleaux triangles. Besides other results, it is also shown in [139] that the densest *I*-packing of a Reuleaux triangle occurs when *I* is the group generated by two rotations of period 3. We refer also to [970], where results on densest packings of Reuleaux triangles are independently re-obtained. Investigating densities of packings in \({\mathbb E}^3\), the authors of [110] used also certain affine images of bodies of constant width, see Section 11.6.

Another interesting problem in which only the case of balls has to be considered is that of finding the largest number of nonoverlapping translates of a given body of constant width \(\Phi \) that can touch \(\Phi \) without intersecting \({{\,\mathrm{\mathrm {int}\,}\,}}\,\Phi \).

We now discuss some results regarding a translative packing problem that cannot be reduced to a corresponding packing of balls. It concerns the

*permeability* of a layer of plane convex figures in

\(\mathbb {E}^2\). For a given figure of constant width

\(\Phi \), let

\(\{\Phi _i\}\) be a translative packing in a strip of width

*t*. Assume that

*t* is minimal for the given collection

\(\{\Phi _i\}\) and let

\(\lambda \) be the infimum of the lengths of all continuous rectifiable curves connecting the two boundary lines without meeting interior points of any

\(\Phi _i\). The

*permeability* of

\(\{\Phi _i\}\) is defined by

$$\begin{aligned} \alpha ( \{\Phi _i\})=\frac{t}{\lambda }. \end{aligned}$$

In [552], Hortobágyi proved that

$$\begin{aligned} \alpha ( \{\Phi _i\})\ge \frac{\sqrt{27}}{2\pi (t-h)+ h\sqrt{27}}, \end{aligned}$$

where

*h* is the width of

\(\Phi \). Equality is possible if

\(\frac{2(t-h)}{\sqrt{3}h}\) is an integer. There are some improvements of this estimate due to Florian [364, 365]; he also studied the possibility of relating permeability estimates of

\(\{\Phi _i\}\) to those for

\(\{\Phi _i^*\}\).

Packing problems of constant width bodies which are not necessarily translative were also studied. A typical problem of this kind concerns the *Newton number*. Given a figure of constant width \(\Phi \), the Newton number \(N(\Phi )\) of \(\Phi \) is the largest number of congruent nonoverlapping copies of \(\Phi \) that touch \(\Phi \). It has been shown by L. Fejes Tóth [344] that \(N(T)\le 7\) for the Reuleaux triangle *T*. Moreover, it has been proved by Schopp [1044] that \(N(\Phi )\le 7\) for any figure of constant width \(\Phi \subset \mathbb {E}^2\). On the other hand, Hortobágyi [551] has shown that any eight figures of the same constant width \(\Phi , \Phi _1,\cdots , \Phi _7\) can always be arranged so that each \(\Phi _i\) touches \(\Phi \) and any two \(\Phi _i\) do not overlap, thus proving that for any plane convex body of constant width \(\Phi \), the largest number of congruent nonoverlapping copies of \(\Phi \) that touch \(\Phi \) is 7. That is, its Newton number \(N(\Phi )=7\).

We come now to *coverings*. A convex set *K* is called a covering set of a class *M* of sets in \({\mathbb E}^2\) if any member of *M* can be covered by a congruent copy of *K*; and *K* is said to be a minimal covering set if it cannot be replaced in this framework by a proper subset. Answering Problem 28 from Chapter VIII of [151], Weissbach [1195] showed that there are sets which are minimal covering sets for the class of all planar sets of constant width, except for closed Reuleaux triangles, thus also suggesting a sharpened reformulation of this problem.

Bezdek and Connelly [116] defined a set *K* to be a *translation cover* for a class *W* of sets if any member of *W* can be covered by a translate of *K*. They showed that every planar set of constant width \(\frac{1}{2}\) is a translation cover for the class of planar closed curves of length 1, and that in fact these are the convex translation covers of minimum perimeter for this class, see Theorem 17.2.4. Their investigations were continued by [766]. In [513], it was proved that a closed plane curve of length \(2 \pi \) can be covered by a rectangle of area 4; if no smaller rectangle will do this, the curve under consideration has to be of constant width 2.

Buchman and Valentine [191] proved that if each \(n+1\) members of a family of convex bodies in \({\mathbb E}^n\) can be covered by some set of constant width 1, say, then all members of the family can be covered this way, see Theorem 15.1.3. Further Helly-type conjectures posed in [191] (see also [272, pp. 131–132]) are investigated in [30]: if, for a given positive number *h*, every \(n+1\) members of a collection of compact convex sets in \({\mathbb E}^n\) simultaneously contain a set of constant width *h*, then all members of this collection contain a set of constant width \(h^*\), where \(h^* = \frac{h}{\sqrt{n}}\) for odd *n*, and \(h^* = 2h - h\sqrt{2n/(n+1)}\) for even *n*. Moreover, an oracle-based algorithm is presented which determines a set of constant width that the members of the given collection have in common. Analogous results for a generalization of the notion of convex sets were derived in [29].

Eggleston [309] proved that any planar convex set of width *h* contains a convex subset of constant width at least \(h/(3-\sqrt{3})\); this bound is sharp for an equilateral triangle. On the other hand, it was proved in [317] that among the largest equilateral triangles which may be inscribed in different types of curves of constant width that one inscribed in a Reuleaux pentagon is the least. In [1094], it is shown that the sum of the diameters of three sets, each of diameter smaller than *h*, which can cover a set of constant width *h*, is larger than 2*h*. The author uses the fact that the perimeter of a triangle whose vertices are from the distinct arcs of Reuleaux triangles of width *d* has to be larger than 2*h*. Besicovitch [106] confirmed partially the following question of Klee: is it possible to inscribe any semicircle of diameter \(h > 0\) in every set of constant width *h*. With restrictive boundary conditions he proved that three semicircles of diameter *h* can be inscribed in a set of constant width *h*. Eggleston [315] continued this by proving that every set of constant width *h* (without any boundary condition) contains a semicircle of diameter *h*. Cooke [264] proved that, again without boundary conditions, there are always at least three distinct such semicircles, and that this is best possible. In a simple geometric way this was reproved by [641], and in [330] this was finally extended to the configurations of plane convex containers of minimal width *h*, covering three semicircles of diameter *h*, see Theorem 5.3.3. Lutwak [741] showed that any closed curve of length 1 can be covered by a semicircular disk of radius \(\frac{1}{\pi }\). Moreover, he showed that this radius cannot be decreased if and only if the curve has the same constant width. This implies that any two closed curves of length 1 can be positioned inside a circle of radius \(\frac{1}{\pi }\) such that their interiors are disjoint. Taking two sets of constant width \(\frac{1}{\pi }\), one sees that the radius of the circle cannot be decreased, and there are no other cases of this kind. As we already mentioned, the article [1192] refers to several interesting results on sets of constant width which are not covered by the survey [238]; these refer to universal covers, to the least number of translates of the interior of a set of constant width required to cover the original set, to cylindrical pipes of minimal radius through which any set of fixed constant width passes, and to equiangular polygons circumscriptible about constant width sets.

For a given family *W* of convex bodies in \({\mathbb E}^n\), let \(a_m(W)\) be the largest number \(\alpha \) such that every set \(A \in W\) contains a polytope *P* with at most \(m > n\) vertices such that the minimal width of *P*, denoted by *w*(*P*), has the lower bound \(\alpha w(A)\). Among many other results, the authors of [451] take *W* also as class of all bodies of constant width, and they obtain lower bounds for \(a_m(W)\), getting, e.g., the Reuleaux pentagon as an extremal figure.

It is well known that the minimal number of directions necessary to *illuminate* the whole boundary of a convex body *K* equals the minimal number *h*(*K*) of smaller homothets of *K* sufficient to cover *K* (see, e.g., [202, Chapter VI]). For this covering problem (often called Hadwiger’s covering problem) also results referring to constant width bodies were derived. For example, it was shown by Lassak [692] that for any 3-dimensional constant width body *K* the inequality \(h(K) \le 6\) holds. The same estimate can be obtained from more general results of Weissbach [1192]: if *K* is an *n*-dimensional body of constant width and \(N(\pi /4)\) denotes the smallest positive integer *m* such that the unit sphere can be covered by *m* congruent caps of spherical radius \(\pi /4\), then \(N(\pi /4)\) is an upper bound for *h*(*K*). Going further with this method, one gets \(h(K) \le \sqrt{2}(1+o(1)), n \rightarrow \infty \), for \(K \subset {\mathbb E}^n\) having constant width. With a more complicated method, Schramm [1047] obtained the upper bound \(h(K) \le 5n \sqrt{n}(4 + \log n)(3/2)^{n/2}\) for constant width bodies *K* in \({\mathbb E}^n\), yielding also \(h(K) \le \sqrt{3/2}(1+o(1)), n \rightarrow \infty \). For small dimensions, the estimate of Weissbach is better, in higher dimensions Schramm’s estimate is much stronger. Weissbach [1193] also studied a modification of the number *h*(*K*) (in terms of illumination), namely suitably referring to all congruent copies of a convex body. He obtained respective upper bounds for constant width bodies in \({\mathbb E}^n\).

**The Borsuk Problem**

We first refer to some basic surveys on *Borsuk’s partition problem* taking also the role of constant width sets into consideration, namely to [488], [272, D 14], [151, Chapter V, and Problems 13–17 in Chapter VIII], [960], [602], and many references therein; see also [238, pp. 79–80].

A part of the information about the history of Borsuk’s problem are already given at the end of Section 15.3. Boltyanski [146] proved that a plane convex figure has the (maximal) Borsuk number 3 if and only if it can be completed to a figure of constant width in a unique way, see Theorem 15.3.3. In [652] it was proved that a planar convex body has a unique completion iff for every of its non-diametral chords (say *ab*) there exists a diametral chord of it whose relative interior is disjoint from *ab*. Lenz [716] proved that the Borsuk number of smooth convex bodies *K* in \({\mathbb E}^n\) is smaller than \(n+1\) if and only if *K* is not of constant width (see also [824] and [501]). Dol’nikov [299] used bodies of constant width to study properties of (geometric) diameter graphs in \({\mathbb E}^3\) and thus to answer the Borsuk problem for finite sets in \({\mathbb E}^3\). To solve the 3-dimensional case, in [962] universal covering systems are used which are based on some kind of Reuleaux polyhedra. For more about translating Borsuk’s problem into the language of graph theory and similar methods, we refer to the survey [961]. Also the paper [300] refers to Borsuk’s problem; one of the numerous related corollaries of that paper concerns the partition of sets of constant width and yields a generalization of Borsuk’s problem. In [532], a special type of Borsuk partition (called cylindrical partition) is studied, where each partition piece can be represented as the intersection of the starting set *K* and some cylinder; it turns out that *K* is of constant width if and only if it does not admit a cylindrical partition. In [652], the following was proved: The Borsuk number \(\beta ({\mathbb E}^n)\) equals \(n+1\) if and only if for all constant width bodies *K* and *M* in \({\mathbb E}^n\) the equality \(\beta (K+M) = \mathbf{min} \{\beta (K), \beta (M)\}\) holds. For *K* a convex body of diameter 2 in \({\mathbb E}^n\), Schulte [1048] showed that a body *C* of constant width 2 containing *K* exists such that every symmetry of *K* is one of *C* and every singular boundary point of *C* is a boundary point of *K* for which the set of antipodes in *C* is the convex hull of the antipodes in *K*. Based on this property he proved Borsuk’s conjecture for convex bodies having no point as endpoint of more than one diameter, see Section 7.5. In [272, A 27], midpoints of diameters of sets *C* of constant width 1 are discussed, based on the papers [825] and [823]: how large can the diameter of the set *M* of all midpoints of diametrical chords of such sets *C* be? If one could always find a simplex containing *M* and being contained in *C*, then the Borsuk problem would have an affirmative answer.

Ivanov [573] described geometric properties of constant width bodies such that the Borsuk number \(n+1\) is guaranteed; see also [574], [575], and [576] for related results and extensions to strictly convex normed spaces. The main result of [1047] on Hadwiger’s covering problem for constant width bodies (see also [1192] and above) yields also an interesting new bound for Borsuk’s partition problem of constant width bodies. For further results and observations regarding Borsuk partitions and constant width bodies we refer also to [651]. The *k*-fold Borsuk number of a set *S* of diameter \(d > 0\) is the smallest integer *m* such that there is a *k*-fold cover of *S* with *m* sets of smaller diameters. Among other results, the authors of [565] characterize the *k*-fold Borsuk numbers of bodies of constant width in \({\mathbb E}^n\).

An interesting covering problem related to the Borsuk conjecture is one in which the symmetry group plays an important role. In this vein Rogers [982] constructed an 8-dimensional body of constant width 1 that is invariant under the symmetry group determined by a regular simplex in \(\mathbb {E}^8\) but cannot be covered by nine convex bodies of diameter less than 1, each having this same symmetry group. This contrasts with the fact, also proved by Rogers, that any *n*-dimensional body of constant width 1 with the symmetry group of the regular simplex can be covered by \(n+1\) sets of diameter less than 1 if there are no symmetry requirements on the covering sets.

Some results related to the Borsuk conjecture in the plane are the following: Let \(\Phi \subset \mathbb {E}^2\) be a figure of constant width 1.

In [824], Melzak proved that

\(\Phi \) can be covered by three subsets of diameter at most

$$\begin{aligned} \mathbf{min} \{ e^+(\Phi ), \sqrt{3}-e^+(\Phi )\}, \end{aligned}$$

where

\(e^+(\Phi )\) is the perimeter of the largest equilateral triangle inscribed in

\(\Phi \). In [822] an upper bound for the diameters of three subsets, into which any planar convex body can be partitioned, is derived. It is given in terms of the edge-length of the largest equilateral triangle whose vertices sit on the boundary of that body, and it improves a result of Gale [386].

Schopp [1045] proved that \(\Phi \) can be covered with three circular disks of diameter \(e^-(\Phi )/6\), where \(e^-(\Phi )\) is the perimeter of the smallest equilateral triangle containing \(\Phi \). Chakerian and Sallee [242] proved that \(\Phi \) can be covered by three translates of any figure of constant width not less than 0.9101.

There are some results that go in the opposite direction by showing that the covering sets cannot be too few or too small. In the case of a figure \(\Phi \) of constant width 1, Lenz [717] proved that at least one of the covering sets must have diameter at least \(\sqrt{3}-1\) and greater than \(\sqrt{3}-1\) if \(\Phi \) is not a Reuleaux triangle.

**Lattice Points **

Let us consider an *n*-dimensional lattice \(\Lambda ^n\) in \(\mathbb {E}^n\), and let us denote by \(\Lambda _0^n\) the lattice of all points of \(\mathbb {E}^n\) with integral coordinates.

Sallee [997] proved that there exists an essentially unique body of maximal constant width \(h_0\) that contains no point of \(\Lambda _0^2\) in its interior. This is a Reuleaux triangle whose width \(h_0\) is a root of a certain polynomial of fourth degree. Numerically, \(h_0\) is between 1.545 and 1.546. Also of interest is the existence of figures of maximal constant width whose interior avoids a locally finite collection of convex sets in \(\mathbb {E}^2\). The resulting maximal sets, if they exist, are certainly Reuleaux polygons. The proof of these results uses an interesting procedure that enables one to construct new Reuleaux polygons from a given one by a replacement procedure for boundary arcs.

The following result, proved by Elkington and Hammer [319] using Sallee’s theorem, is related. Let

*g*(

*c*) denote the minimum of the number of points of

\(\Lambda _0^2\) in a figure of constant width greater than

*c*. Then

$$\begin{aligned} (\frac{c}{h_0})^2 \le g(c) \le \frac{c^2}{2}(\pi -\sqrt{3}). \end{aligned}$$

Hence, if one is interested in densest lattice packings of

\(\Phi \) (that is, packings of the form

\(\{\Phi +p\mid p\in \Lambda ^n\}\)) it follows from (

15.2) that the direction of one of the basic vectors of

\(\Lambda ^n\) may be selected arbitrarily. L. Fejes Tóth [343] noticed this property of sets of constant width (when

\(n=2\)) and proposed to characterize all convex bodies that permit such direction-invariant densest packings.

A lattice in the plane is called a *holding-lattice of a planar set* *S* if any set congruent to *S* contains at least one lattice point; if its fundamental triangle has the greatest possible area, the lattice is called “thinnest”. In [111] it is proved that, also for certain types of constant width sets, the thinnest holding-lattice is based on an equilateral triangle. Let \(P \subset {\mathbb E}^2\) be a convex polygon, and *K* be a planar convex body. Peri [925] calculated explicitly the measure of all sets congruent to *K* which can be contained in *P* in the cases when *P* is a parallelogram and *K* is either a constant width set or a regular convex polygon. Identifying *P* as the fundamental region of a lattice, these results are applied to corresponding problems in geometric probability. Similarly, in [10] the measure of families of congruent constant width bodies in \({\mathbb E}^3\) is computed which, under certain conditions, are entirely contained in a rectangular parallelepiped or in a tetrahedron. Based on this one can compute the probability that such a body, placed uniformly at random in the space of a lattice based on parallelepipeds or regular tetrahedra, meets the planes of the lattices. The paper [211] refers to an extension of Buffon’s needle problem, namely taking (instead of segments) convex test bodies of given perimeter and diameter in the Euclidean plane in view of a lattice of lines. The cases of ellipses and constant width sets are also studied. Similarly, in [17] two events are taken into consideration, namely that a random congruent copy of a convex body in the plane meets each of two given families of equidistant lines, with angle \(\theta \) between them. It is shown that there is always a value of \(\theta \) such that the two events are independent. Moreover, they are independent for any \(\theta \) iff the test body is of constant width. Here we mention also the paper [547]. Also (but differently) connected with the concept of distinct families of parallel lines and planar convex bodies is the paper [303]. Given such a body *K* and a sweep line as a tool, one might ask for the best procedure to reduce *K* by a sequence of sweeps to a point. The authors consider a variant of this problem, for which also constant width sets play a role. Also in [547] results connecting Buffon’s needle problem and sets of constant width are announced.

**In- and Circumscribed Sets**

Clearly, some of the covering results discussed above might be also interpreted as *inscription-circumscription results*, such as those with inscribed semicircles. It is a question of personal judgement where they should be located in our framework.

Regarding regular simplices which are in- and circumscribed about bodies of constant width we refer the reader to a nice and comprehensive discussion on page 74 of [238]. Specializing the results presented there, it was first observed in [309] that an equilateral triangle of minimal width \(\Delta \) contains no set of constant width larger than \(\Delta (3-\sqrt{3})\). See also [311, pp. 140–149], where it is furthermore shown that any plane convex body of minimal width \(\Delta \) contains a set whose constant width equals \(\Delta (3-\sqrt{3})\). Using suitable intersections of regular simplices *S* and \(-S\) as polytopes circumscribed about bodies of constant width, one gets for \(n = 2\) the result of Pál [909] that each plane set of constant width admits a regular circumscribed hexagon (see also [160, p. 131]), and for \(n = 3\) the observation of Gale [386] that any 3-dimensional body of constant width has a circumscribed regular octahedron, see also the proof of Theorem 15.2.5. Pál’s theorem was generalized in [250]: If the width function of a strictly convex body \(K \subset {\mathbb E}^2\) has period \(\frac{\pi }{3}\), then there is a regular hexagon circumscribing *K*. Clearly, if some polytope *P* circumscribes every convex body of constant width *h*, it contains every set of diameter *h*. Motivated by the facet number \(n(n+1)\), which is the largest possible for the circumscribing property, Makeev [758] conjectured that every constant width body in \({\mathbb E}^n\) is inscribed in a polytope similar to the dual of the difference body of a regular *n*-simplex. In [664], Makeev’s conjecture was confirmed for \(n = 3\), and also further related results were obtained there, see Section 16.3.

A convex body in \({\mathbb E}^n, n > 2,\) is of constant width if and only if all its circumscribed boxes are mutually congruent cubes. This is not true for \(n = 2\) (already the square itself is such a set). Such sets are investigated in [1204, p. 97], and polygonal versions are studied in [607] and [608]. While all circumscribed boxes of an *n*-dimensional constant width body have the same volume, the converse is not true. Petty and McKinney [934] investigated this family of bodies, and in [1204, p. 93] a detailed representation of plane convex sets with constant perimeter of their circumscribed rectangles is given. Characterizing affine images of *n*-dimensional constant width bodies, \(n\ge 3\), Chakerian studied in [229] convex sets with the property that all their circumscribed boxes have constant diagonal lengths, see Theorem 11.6.3. Toranzos [1133] characterized sets of constant width in terms of circumscribed rhombuses by the following property: if three sides of a rhombus span supporting lines of a plane curve, then also the fourth. (If “rhombus” is replaced by the union of two isosceles triangles, this condition characterizes circles.)

For convex figures in the plane, De Valcourt (see [293], [294], and [295]) defined measures of axial symmetry by means of different types of inscribed figures; he paid special attention to the situation when the given figures are of constant width, and he posed several interesting questions.

**Further Topics**

Another interesting topic from the combinatorial geometry of convex bodies, which can be described in terms of illumination systems, is that of *primitive fixing systems* of a convex body *K* in \({\mathbb E}^n\) (see [151, § 44]). Such a point system *F* from the boundary of *K* will stabilize *K* against translations, and no proper subset of *F* does the same. If in this procedure frictions are not allowed, then we call this variation of *F* a *primitive hindering system* of *K*. The papers [148] and [149] contain also results on the cardinalities of maximal primitive fixing and hindering systems of plane sets constant width. For example, in [149] it is shown that for such sets the maximal cardinality of a primitive hindering system equals 3 for Reuleaux triangles, 5 for Reuleaux pentagons, and 4 otherwise.

A further variation of illuminating convex bodies *K* in \({\mathbb E}^n\) is that of *X*-raying them. The *X*-*ray number* *X*(*K*) is defined as the smallest number of lines \(\{L_i\}\) such that each point \(p \in K\) is contained in some line parallel to one element of \(\{L_i\}\) and intersecting the interior of *K*. The authors of [119] proved that for *K* a constant width body, \(X(K) \le 6\) holds for \(n=4\), and \(X(K) \le 2^{n-1}\) holds for \(n=5\) as well as \(n=6\).