**The Blaschke–Lebesgue Problem**

The Blaschke–Lebesgue theorem asserts that among all plane convex sets of the same constant width only the Reuleaux triangle has the minimum area, see Theorem 12.1.5. In [160, pp. 132–133], [202], and [137] the proof of Lebesgue is reproduced, and references to proofs of Blaschke and Fujiwara are given. A variant of Lebesgue’s proof can be found in [1204]. Blaschke’s proof of the minimum property of the Reuleaux triangle (see [131] and, for older references to this topic, [598]) consists of two steps: First it is proved for Reuleaux polygons (by a geometric method developed by Steiner to prove the isoperimetric property of the circle), and then it is verified for general constant width sets via an approximation step (a variational problem with curvature restrictions). We refer to [667] for a more rigorous modification of Blaschke’s approach (all necessary lemmas are rigorously proved, thus filling gaps in Blaschke’s proof). Further on, the modifications yield a purely geometric proof of the Blaschke–Lebesgue theorem (avoiding any compactness arguments). An early analytical proof of the Blaschke–Lebesgue theorem was given by Fujiwara (see [382] and [383]), and more recently Harrell [515] presented the result also in analytic form, posing it as a variational problem for the radius of the curvature function. This admits a direct solution yielding both the uniqueness (up to rigid motions) and the structure of the minimizing set. A proof by Ghandehari [412] using Pontryagin’s maximum principle has a few points in common with the arguments from [515], and also with the determination of the *m*-orbiforms of least area that can be rotated inside a given regular *m*-gon in [636]. Here we also mention [765], for a related formulation of the Blaschke–Lebesgue theorem in terms of optimal control theory. In [84], methods from semidefinite programming are used to attack the Blaschke–Lebesgue problem in the plane; see also the survey [85].

Ohmann [898] gave a direct generalization of Lebesgue’s theorem regarding the least area of the Reuleaux triangle among all sets of the same constant width. His approach uses affine regular hexagons inscribed to plane curves, and the result can be interpreted as an analogue of Lebesgue’s theorem for normed planes.

Mayer [804] gave a broad discussion of properties of constant width sets and the Blaschke–Lebesgue problem, using notions like isoperimetric deficit (which, as area quotient of the considered set and a circle of equal perimeter, can also be interpreted as a measure of asymmetry). There are many further proofs of the Blaschke–Lebesgue theorem (see, e.g., [307], [308], [106], [230], [209], [667], and [1203]), and further elementary ones can be found in [312] and [105]. Using mixed areas, Chakerian [230] found also a short proof, allowing the extension to normed planes. See the proof of Theorem 12.1.5. In [1218], the Blaschke–Lebesgue theorem is confirmed for a special class of constant width figures.

In [458] (see also the survey [460, § 6.2]), various stability results referring to the Blaschke–Lebesgue theorem are established. For example, if *C* is of constant width 1 and area at most \(A_0 + \varepsilon \) (where \(A_0\) denotes the area of the Reuleaux triangle of width 1), then there exists a Reuleaux triangle whose Hausdorff distance from *C* is at most \(\sqrt{10 \varepsilon }\).

In [123], the following extensions of the Blaschke–Lebesgue theorem are discussed: A disk-polygon is the intersection of a finite set of disks of radius 1 in the Euclidean plane. For a parameter \(0< d < \sqrt{3}\) denote by *F*(*d*) the class of disk polygons such that the distance between any two centers is at most *d*. Denote by \(\triangle (d)\) the regular disk triangle whose three generating unit disks are centered at the vertices of a regular triangle of side length *d*, where \(1 \le d < \sqrt{3}\). The author proves that \(\triangle (d)\) is minimal in *F*(*d*) regarding area, inradius, and width.

Staying with the Blaschke–Lebesgue theorem and related topics, we now turn to concrete inequalities. For a plane set

*C* of constant width

*h* and of area

*A*(

*C*) the Blaschke–Lebesgue theorem is equivalent to the inequality (see Section

14.1)

$$ A(C) \ge \frac{\pi - \sqrt{3}}{2} h^2\,, $$

which can also be used in normed planes if in such planes a suitably “generalized Reuleaux triangle” is analogously constructed (see Chapter

10 and [898], [661]). The latter paper contains a more general inequality for convex figures, estimating their area against minimal width and diameter and characterizing generalized Reuleaux triangles. If the diameter and the minimal width coincide, then the inequality above is obtained. Sholander [1063] adds the perimeter to these three quantities and considers triples of quantities, in each case leaving two of them fixed and extremizing the third; see also [1019] for such complete systems of inequalities and the short discussion of Blaschke–Santaló diagrams below (see also [177]).

A modified Blaschke–Lebesgue problem is investigated in [912]: the authors ask for plane convex sets *C* of constant biwidth having minimal area (the biwidth of *C* in direction *u* is in fact the sum of two widths of *C* - that in direction *u* and that in direction \(u + \frac{\pi }{m}\)). When *m* is an odd integer, the Reuleaux triangle is extremal, and for the even case certain types of Reuleaux polygons have the least area. The idea of considering the quantity “width” as sum of widths (or support values) in different directions goes back perhaps to [905], see also [1219] for properties of the related sets.

Eggleston [314] showed that the width of a constant width set *C* in the plane that contains a given square *S* largest when *C* is a Reuleaux triangle, and he gives a respective bound for the edge-length of *S*. In [516], the concept of *m*-*diameter* of a set *S* is considered. This quantity is defined as the supremum of the geometric mean of all Euclidean distances among *m* (and not “only” two, as in the usual way) points from *S*. In this paper, it is shown that among all sets of constant width the Reuleaux triangle has the largest (and the circle the smallest) 3-diameter. It is proved that there are infinitely many non-circular curves of constant 3-diameter. In [570], analogous results for \(m=5\) and, with additional assumptions, for \(m=7\) are derived. Another concept of “*m*-width” was considered in the papers [252] and [447].

In some sense “dual” to the Blaschke–Lebesgue theorem, but restricted only to Reuleaux polygons, is the following analogue of the classical Zenodorus problem for polygons: Which Reuleaux *m*-gons of given width have maximal area? Firey [355] characterized the regular Reuleaux *m*-gons by this property, and his approach to this result and related statements was simplified by Sallee [999], therefore yielding the name “Firey–Sallee theorem” for this result. The authors of [667] used a modification of Blaschke’s original approach to the Blaschke–Lebesgue theorem for presenting also a rigorous proof of the Firey–Sallee theorem.

We also mention that the Blaschke–Lebesgue theorem plays a role for studying upper bounds on the expected value of the number of normals passing through a point randomly chosen from a convex body, see [301].

**The Blaschke–Lebesgue Conjecture**

We continue with the 3-dimensional case of the Blaschke–Lebesgue problem. Here the volume-minimizing procedure is clearly equivalent to minimizing surface area, due to Blaschke’s theorem on volume and surface area in that dimension, see Theorem 12.1.4. It is a long-standing conjecture that the 3-dimensional Meissner bodies (see, e.g., [160, pp. 135–136], [1204, § 7], [272, A. 22], [151, Chapter VIII], and [612]) are the minimizers. In [610], Kawohl mentioned that one of his students generated randomly one million of 3-dimensional bodies of constant width—none of them had smaller volume than the Meissner bodies. Assuming that this conjecture were correct, the minimum volume of a body of constant width 1 in \({\mathbb E}^3\) would be about 0.42. Chakerian [230] confirmed a lower bound of roughly 0.365 (see Theorem 14.1.2); see also [356]. Also, the paper [494] is devoted to the 3-dimensional situation. The comparison of the volume of a constant width body with that of the ball having the same width yields a functional which is proved to increase when going from a body to its parallel body. i.e., for improvement one has to shrink the boundary along the normals. To keep convexity, this process has to be stopped when the boundary touches the set of foci (for rational support functions explicit examples of constant width are constructed). In [31], a necessary condition in this direction is given: For a local minimizer of the Blaschke–Lebesgue problem in \({\mathbb E}^3\) having constant width *h*, the smooth parts of its boundary have their smaller principal curvature constant and equal to \(\frac{1}{h}\) (see Theorem 14.2.1). Hence the smooth boundary parts are spherical caps or pieces of tubes, each of radius *h* (see Corollary 14.2.1). In [86], 3-dimensional constant width bodies are presented analytically, using a bijection between spaces of functions and constant width bodies (see also [31] and [677]). The authors compute several quantities (like volume or surface area) of them. The paper concludes with a necessary condition for bodies of given constant width to have the minimal volume. The Blaschke–Lebesgue theorem plays also a role for determining upper bounds on the number of normals through randomly choosen points of constant width sets, see again [301]. Also in [123] the Blaschke–Lebesgue theorem is discussed for three dimensions. Campi et al. [209] showed that the restriction of the Blaschke–Lebesgue problem to 3-dimensional bodies of revolution yields the body obtained by rotating the Reuleaux triangle about one of its symmetry axes as extremum (see Theorem 14.2.2).

Going to higher dimensions, results of Firey [356] and Chakerian [230] were improved (for

\(n > 4\)) by Schramm [1046], see also the discussion in [272, p. 34]. He proved that for a body

*C* of constant width 2 in

\({\mathbb E}^n\) and

*B* as unit ball the lower bound

$$ V(C) \ge \left( \sqrt{3+2/(n+1)}-1\right) ^n \cdot V(B) $$

for the volume

*V*(

*C*) holds. These investigations on volume-minimizers led Heil [524] to define the fruitful notion of

*reduced bodies* (see Chapter

7); see also other related problems of Heil in pp. 260–261 of the problem collection [481] and the conjecture of Danzer formulated there, namely that the searched volume-minimizers have the symmetry group of a regular simplex. Here we also mention that Firey [356] derived inequalities for bodies of constant width also holding with respect to the minimal brightness (= minimal volume of orthogonal (

\(n-1\))-projections).

**Measures of Asymmetry**

Before we start to discuss results on asymmetry measures (which can be defined in various ways), we refer to Grünbaum’s excellent related survey [487], nicely presenting the basic notions and the state of the art at that time, see also [272, A 15]. A recent comprehensive reference on measures of asymmetry is the book [1135]; Chapter 2 there is dedicated to bodies of constant width and related topics, such as Klee’s critical sets and affine diameters. In contrast to the general point of view presented in [487], we restrict ourselves here to bodies of constant width.

To start with a suggestive example of a very natural measure of asymmetry for constant width sets, we present one studied by Besicovitch [103] for the planar case: Let *C* be a planar closed convex curve, and \(\alpha \) be the ratio of the maximal area of a centrally symmetric closed curve contained in *C* and of the area enclosed by *C* itself. Then \(\alpha \) is a measure of asymmetry of *C* in the sense of [457], and the smallest value \(\frac{2}{3}\) is known to be reached exactly by triangles. In [103], this consideration is restricted to the class of constant width curves, and it turns out that the largest asymmetry (i.e., the smallest ratio \(\alpha \)) is attained precisely by the Reuleaux triangle. The exact value of \(\alpha \) (roughly about 0.84) is derived, too. We note that Estermann [322] introduced the “inverse outer measure”, defined via a centrally symmetric convex body of minimal volume containing a fixed convex body; its restriction to bodies of constant width was investigated by Chakerian [230]. Another suggestive example is presented in [465]: For each direction *u*, a plane constant width set *C* has a unique diameter splitting the area of *C* into two subsets of areas \(A^+(u)\) and \(A^-(u)\), respectively. The authors define the asymmetry function as the maximum of the ratio \(A^+(u)/A^-(u)\), and they show that it lies between 1, characterizing the circle, and \((4\pi - 3 \sqrt{3})/(2\pi - 3 \sqrt{3})\), characterizing the Reuleaux triangle.

Continuing studies of Besicovitch’s concept, Eggleston [307] defined that for a number *k* with \(\frac{1}{2} \le k \le 1\), \(R_k\) denotes the set of points at distance \(\le 1-k\) from the Reuleaux triangle of width \(2k-1\). He showed that for each set *C* of constant width 1, such that any boundary point of it has curvature between *k* and \(1-k\), the relation \(\alpha (C) \le \alpha (R_k)\) holds. Chakerian and Stein [243] introduced an asymmetry measure for plane convex bodies which is defined with the help of areas created by points which occur as midpoints of a fixed number of chords passing through them. They investigated this measure also for constant width sets, getting analogously the Reuleaux triangle and the circle as extremal cases.

Also in [590], [465], and [736], extremal planar constant width figures for several related measures of asymmetry are discussed, analogously yielding Reuleaux triangles as most asymmetric constant width figures. For example, in [736] the authors refer to [465] (see above), but replacing the area by perimeter. Related investigations on Reuleaux polygons are presented in [495] and [589].

Results on asymmetry measures for higher dimensions were obtained in [591] and [592]. For

*K* a convex body in

\({\mathbb E}^n\) and

*H* a hyperplane passing through an interior point

*x* of

*K*, let

\(\gamma (H, x)\) be the ratio, not less than 1, in which

*H* divides the distance between the two supporting hyperplanes of

*K* parallel to

*H*. Then the Minkowski measure of asymmetry of

*K* is defined by (see Section

14.3)

$$ \mathrm{as}\, (K) : = \mathbf{min}_{x \in \mathrm{int} K} \mathbf{max}_{x \in H} \gamma (H, x)\,. $$

In [591], the following inequalities for this measure of asymmetry are proved:

$$ 1 \le \mathrm{as}\,(K) \le \frac{n+\sqrt{2n(n+1)}}{n+2}\,. $$

Equality on the left-hand side holds iff

*K* is an Euclidean ball, and if

\(n=3\), equality holds on the right-hand side if

*K* is a Meissner body. Continuing this, it is shown in [592] that the upper bound is reached by bodies of constant width which are completions of a regular

*n*-dimensional simplex, i.e., a complete convex body containing a regular

*n*-dimensional simplex and having the same diameter as the simplex. In [587], Besicovitch’s result (that the most asymmetric constant width set in the plane is the Reuleaux triangle regarding

\(\alpha \)) is reproved, and in [588] the same author applied his methods to measures of asymmetry of rotational constant width bodies. Continuing the work [465], also in [913] asymmetry measures for constant width bodies of revolution in

\({\mathbb E}^3\) are studied; the body obtained from a Reuleaux triangle via rotation about one of its symmetry axes is shown to be extremal.

The so-called mean Minkowski measures of convex bodies (introduced by Toth, see [1135]) were investigated for the class of constant width bodies in [593]. With respect to these measures, completions of regular simplices are the most asymmetric bodies.

In [294], for the class of plane convex bodies *K* a “measure of axial symmetry” is defined as a real-valued function *f* satisfying \(0 \le f(K) \le 1\), with \(f(K)=1\) iff *K* is symmetric with respect to some line, and having similarity-invariance (see also [293]). Eleven such measures of axial symmetry are studied in [294], bounds for them are established (also for the case when *K* ranges over all sets of constant width), and interesting questions are posed.

**In- and Circumradii**

A well known property of

*n*-dimensional bodies of constant width

*h* is the concentricity of their circumsphere and insphere. The respective in- and circumradius,

*r* and

*R*, satisfy the relations

$$ r+R = h \, \text{ and } \, h\left( 1-\sqrt{\frac{n}{2n+2}}\right) \le r \le R \le h \sqrt{\frac{2}{2n+2}} \,, $$

cf. Eggleston [308] and Melzak [825] (see also Section

3.4). The planar version of the left equation goes back to Minoda [840], who applied it also to some inequalities for areas. Chakerian [231] extended this relation to normed spaces, see Chapter

10.

The union of all points between circum- and insphere of a constant width set *C* is called the minimal shell (for \(n=2\) minimal annulus) of *C*, and in [160] pp. 134–135, references are given for results to the following problem for \(n=2\): given a suitable shell *S*, where *r* and *R* satisfy the relations above, determine those constant width sets which have *S* as their shell and maximal/minimal area. The case of maximal area was solved (see [159] and [160], pp. 134–135), and Mayer [804] gave corresponding upper and lower bounds for areas from which also the Blaschke–Lebesgue inequality above follows. In [805], he presented a sketch of a proof that the minimal area is reached by certain Reuleaux-type polygons.

We continue with the circumradius

*R*, combining it also with other geometric quantities. For a convex body

*K* in

\({\mathbb E}^n\), we write

*D* and

*p* for diameter and perimeter, respectively. Scott [1052] showed that for any convex body

*K* there is a constant width body

*C* containing

*K* and having the same circumradius and diameter as

*K* (see Section

14.4 and also Chapter

7 about completions). Let

\(r'\) and

\(p'\) correspondingly denote the inradius and perimeter of

*C*, related to

*K* in the above sense. Scott used properties of this constant width set

*C* and several combinations of inequalities to obtain new inequalities particularly involving the factor

\(2R-D\), like for example

$$ (2R-D) p \le \left( 2\sqrt{3}-3\right) \pi R^2\,, $$

where equality holds only for Reuleaux triangles. Using

\(\triangle \) for minimum width, this inequality yields

$$ (2R-D) \triangle \le \left( 2\sqrt{3}-3\right) R^2\,, $$

obtained for all plane convex bodies already in [1051], and again with equality only for Reuleaux triangles. In [1052], the latter inequality was extended to

\({\mathbb E}^n\) in the following form:

$$ (2R-D) \triangle \le \frac{2\sqrt{n+1}}{n} \left( \sqrt{2n} - \sqrt{n+1}\right) R^2\,. $$

In [536], it is proved that constant width bodies in

\({\mathbb E}^n\) maximize the minimal width when their circum- and inradius are prescribed; also the paper [535] should be mentioned here.

In [436], the notions of classical radii are extended for bodies of constant width to inner and outer successive radii, and respective inequalities are given. The paper [176] contains a deep study of isoradial bodies which can be seen as a generalization of constant width bodies. Namely, the so-called strongly isoradial bodies are those of constant inner and outer constant *j*-radii, in the plane identical with the family of constant width sets.

The paper [50] refers to *m*-gons inscribed to a curve *C* of constant width *h*. Given a sequence of points \(p_0,p_1, \dots , p_{n+1}=p_0\) in *C*, the author shows how closely *h* and the in- and circumradius of *C* can be approximated.

**Isoperimetric Inequalities and Related Topics**

Combining Bonnesen’s sharpening of the isoperimetric inequality for planar convex figures

*K* (see [160, p. 83]) with relations between quantities of

*K* and of its central symmetral

\(K^* = \frac{1}{2}(K + (-K))\), Ganapathi [391] derived the inequality

$$ p^2 - 4 \pi A \ge \pi (D-\triangle )^2 + 4 \pi (A^* - A)\,, $$

where

\(A^*\) denotes the area of

\(K^*\), with equality only for constant width sets, see also Theorem

14.5.1. The quantity

\((A^*-A)/2\) is interpreted as the area associated with the locus of all midpoints of diametral chords of

*K*, which is maximized by the Reuleaux triangle among all sets of the same constant width (due to the Blaschke–Lebesgue theorem). This inequality was investigated further by Vincensini [1152]. Replacing perimeter by total mean curvature and area by surface area, Ganapathi [392] gave an analogue to the above inequality for the 3-dimensional situation. The value of the right-hand side then becomes proportional to the signed surface area of the midpoint locus of all diametral chords of the convex body under consideration, and again equality holds only for constant width bodies. Groemer studied isoperimetric and related inequalities in Section 4.3 of his book [464], involving also constant width sets. His motivation there is to demonstrate how Fourier series can be used to get stability estimates.

Taylor [1120] found the area minimizers among those plane convex bodies of given perimeter which are contained in a fixed Reuleaux triangle. On the other hand, Minoda [839] searched perimeter minimizers among plane closed curves in which a given convex figure can be rotated (see also Section 17.1). Besides the circle, he obtained in some cases certain non-circular constant width figures as such minimizers.

In [1225], a sharpened isoperimetric inequality for closed, regular convex curves using the oriented area of their Wigner caustics (i.e., certain affine equidistants) is presented. Equality holds exactly if the original curves are of constant width. A related stability result (also for constant width curves) is given, too.

In [224], an inequality between the area enclosed by a plane constant width curve (containing the origin) and the squared length of its pedal curve is obtained, characterizing the circle. In this way, also the isoperimetric deficit of the pedal curve can be estimated.

Clearly, the inequality \(p(K) \le \pi D(K)\) for \(n=2\) implies that for fixed perimeter the diameter *D*(*K*) is minimized if *K* is of constant width. This is a special case of results derived by Sachs [996]. He proved extremal properties of certain types of closed curves via integral power means of lengths of chords varying independently between points of these curves, and among the studied types of curves also constant width curves play a role.

**Further Quantities and Concepts**

The classical Rosenthal–Szasz theorem (see [987] and Section 44 of [160]) says that for a compact, convex figure *K* in the Euclidean plane with perimeter *p*(*K*) and diameter *D*(*K*) the inequality \(p(K) \le \pi D(K)\) holds, with equality iff *K* is of constant width *D*(*K*). In [65] this was also confirmed for Radon planes, and a similar statement (involving the antinorm appropriately) was proved there for all normed planes.

For a compact set *S* in \({\mathbb E}^n\), let \(\beta (S, x)\) denote the maximum of distances from *x* to points of *S*. Reidemeister [964] showed that a convex body *K* in \({\mathbb E}^n\) is of constant width if \(\beta (K, x)\) is constant for *x* running through the boundary of *K*, see Theorem 7.2.3. The following nice conjecture is due to Alexander [20] (see Section 4.3): if *C* is a closed and rectifiable curve in \({\mathbb E}^n\) such that \(\beta (C, x) \ge \lambda \) for all *x* from *C*, then the length of this curve is at least \(\lambda \pi \), and it can attain this length only if it is a plane curve of constant width. This was independently proved by Sallee [1000] and Falconer [329]. Inspired by a conjecture of Herda (see [533] for a related survey), Chakerian [234] proved the following (which is more): if *C* is rectifiable simple closed curve in the plane, and \(x \rightarrow f(x)\) with \(x \in C\) is any involution of *C* without fixed points such that the distance from *x* to *f*(*x*) is at least \(\lambda \) for all \(x \in C\), then the length of *C* is at least \(\lambda \pi \), and equality holds iff *C* is of constant width and all chords connecting *x* and *f*(*x*) are diametral ones. (Since \(\beta (C, x) \ge \lambda \) for all \(x \in C\), Alexander’s conjecture is more general.) Lutwak [740] extended Herda’s conjecture to integral power means of the perimeter bisectors and the area bisectors of convex curves. In [583], a theorem on lengths of chords of constant width curves which connect touching points of circumscribed squares is proved, together with a characterization of the circle. This verifies a conjecture of Green (related to is optics of given curves) at least for this class of convex curves.

Kubota [660] proved the following theorem referring to four quantities: If *C* is a convex figure in the plane with *A*, *p*, *D* as its area, length, and diameter, respectively, *M* denoting the mixed area of *C* and \(C'\) (the figure obtained from *C* by rotation about \(180^\circ \)), then \(Dp \ge 2(A+M)\), with equality only for constant width curves. This brings us closer to topics related to complete systems of inequalities and the Blaschke–Santaló diagram; see, e.g., [1019], [536], [535], and [177]. E.g., in [177] a complete 3-dimensional Blaschke–Santaló diagram for plane convex bodies with respect to their four classical quantities in- and circumradius, diameter, and minimum width is presented. The large variety of extremal planar figures in this framework shows that many of them are somehow (regarding shape) close to the Reuleaux triangle, and several slight modifications of it.

We continue with the same motivation (namely, that shapes of constant width figures play the key role). Again the used constant width figures are not themselves extremal figures, but can help to describe such extremal figures geometrically. This is also due to the fact that most of the questions that we discuss now refer to proper polygons. For example, area-minimal plane convex figures with prescribed diameter and perimeter were studied in [656] and [657]. This yields concave maximum problems, having a certain non-regular inpolygon of the Reuleaux triangle as area-minimal extremum. In [417], suitable circumscriptions of equiangular *m*-gons about constant width sets are investigated, where these *m*-gons should have extremal perimeters; it turns out that smallest and largest perimeters are attained when the used constant width sets are regular Reuleaux *m*-gons. More generally, in the surveys [45] and [47] problems of the following type are presented: let \(P_m\) be a convex polygon with *m* sides, and take its perimeter, diameter, area, sum of distances between the vertices, and minimum width. One is asked to minimize or maximize one of these quantities, while the others are fixed. It turns out that for solving some of these questions, polygonal shapes close to those of Reuleaux polygons (the authors also use the term “clipped Reuleaux polygons”) and diameter graphs of Reuleaux polygons play a key role. Papers from this research direction, in which geometric ideas related to Reuleaux polygons are particularly important, are [280], [109], [373], [46], and [869]. For example, Datta [280] gave a sharp upper bound for the maximum perimeter of a convex *m*-gon of diameter 1 in the plane. The explicit constructions of all extremal *m*-gons show that all of them have equal sides and are inscribed in a Reuleaux *k*-gon, for an odd integer *k* not larger than *m*. Gritzmann and Lassak [451] derived estimates for the thickness of polytopes inscribed to convex bodies, the latter being particularly also centrally symmetric or of constant width. The paper [899] contains a theorem about polygons with prescribed vertex angles, circumscribed about constant width curves and being extremal regarding their perimeters.

Let

*K* be a convex body in

\({\mathbb E}^n\),

*M*(

*K*) its mean width,

\(P_K\) the parallelepiped of minimal mean width circumscribed about

*K*, and

\(C^n\) a cube circumscribed about the unit ball of

\({\mathbb E}^n\). Among various other results, in [164] the inequality

$$ M(P_K) \le \frac{1}{2} M(K) \cdot M(C^n) $$

was established, with equality iff

*K* is of constant width. And every parallelepiped of minimal mean width that is circumscribed about a constant width body is a cube. While all rectangular parallelepipeds circumscribed about a body of constant width have the same volume, this property does not characterize constant width sets; nonspherical bodies satisfying this property may even be centrally symmetric. Petty and McKinney [934] characterized them for

\(n > 2\), and in the plane the analogue of this characterization is still an unsettled problem. Inspired by a conjecture of Moser (on covering closed curves of certain length by rectangles, see D 18 in [272]), the authors of [1027], [241], [513], and [739] considered inequalities referring also to volumes of boxes circumscribed about convex bodies, with cases of equality only for bodies of constant width. For example, it is proved that a closed plane curve of length

\(2 \pi \) can be covered by a rectangle of area 4. If no smaller rectangle suffices, then the covered curves have to be of constant width 2 (this is also related to [1187]). Chakerian and Logothetti [239] proved that the smallest regular

*m*-gon (

\(m > 3\)) which can cover any plane set of diameter 1 is the smallest regular

*m*-gon that can be circumscribed about a Reuleaux triangle of width 1. Bezdek and Connelly [116] showed that any plane closed curve of length 2 can be covered by a translate of the body

*C* if

*C* is of constant width 1, and that, conversely,

*C* must have constant width 1 if its perimeter is not larger than

\(\pi \) (see Theorem

17.2.4). Somehow converse to a result in [317], where largest equilateral triangles contained in constant width sets are investigated, Eggleston proved in [309] that every plane set of constant width

*h* contains a subset of constant width

\(\ge h/(3-\sqrt{3})\), this bound being attained for the equilateral triangle. In [294], the author studies inequalities referring to area and diameter of convex figures

*K* and certain types of axially symmetric polygons inscribed to them; also the subcase that

*K* is of constant width is taken care of.

The approach to the following inequality is nicely presented in Section 7 of [238], and already Kubota [660] verified the right-hand side for

\(n=2\). Namely, for

*K* an

*n*-dimensional convex body,

*s*(

*K*) its surface area, and

\(V(K-K,K,\dots , K)\) symbolizing mixed volume, we have

$$ \frac{1}{n} \triangle (K) s(K) \le V(K-K, K,\dots , K) \le \frac{1}{n} D(K) s(K)\,, $$

where

*V*(

*K*) is the volume of

*K*, and equality holds iff

*K* has constant width, see Section

12.4. Minkowski’s inequalities for mixed volumes yield

$$ V(K) \le \frac{1}{2n} D(K) s(K)\,, $$

true for any convex body and obtained by Firey [356] (equality holds iff

*K* is a ball). See also Theorem

12.4.1. Certain combinations lead to an inequality of Petermann [926], namely

$$ \triangle (K) V(K) \le D(K) V(L, K, \dots , K)\,, $$

where

*K* (being centrally symmetric) and

*L* are convex bodies in

\({\mathbb E}^n\). Equality holds iff

*L* is of constant width and

*K* is a ball. If the body

*K* above is of constant width

*h*, we get

\(V(K)/s(K) \ge h/2n\). If

\(K^*\) denotes again the central symmetral of

*K*, we see from this that if

*K* is a set of constant width, then

$$ V(K)/s(K) \le V(K^*)/s(K^*)\,, $$

as observed by Heil in [481], p. 260.

For mixed volumes of constant width sets

*K* in

\({\mathbb E}^n\), Godbersen [422] verified

$$ V(K,\dots , K,-K,\dots ,-K) \le {\left( {\begin{array}{c}n\\ k\end{array}}\right) } V(K)\,, $$

where

\(k=0,1,\dots , n\),

*K* appears

*k* times and

\(-K\) appears

\(n-k\) times within the mixed volume

\(V(K,\dots , K,-K,\dots ,-K)\) (see also Section

12.4). If this would hold for all convex bodies, the famous difference-body inequality (see, e.g., § 7.3 in [1039]) would directly follow from it. Also, the paper [411] deals with a variety of inequalities involving mixed volumes of a convex body

*K* and its polar body; these results are applied to obtain inequalities for quermassintegrals of

*K* and its polar in case that

*K* has constant width.

Leichtweiss [710] characterized affine classes of convex bodies containing a body of constant width in view of minimal diameter-thickness ratios. In [742], a special case of the Blaschke–Santaló inequality is shown to be equivalent to a power-mean inequality involving the diameters and widths of a convex body *K* in \({\mathbb E}^n\). This leads to strengthened versions of known inequalities and a characterization of the affine images of bodies of constant width, see Section 11.6. In [226], a lower bound on the volume product of a constant width set and its *p*-centroid body (see [1039, § 7.4]) was obtained.

Of course, various inequalities between geometric quantities of constant width bodies give rise to study also stability versions of them; see, e.g., [458] and [460], and we refer also to the subsection above where the Blaschke–Lebesgue theorem is discussed.

Minkowski’s famous theorem about constant perimeters in \({\mathbb E}^3\) (see Theorem 13.1.1), yielding constant width, has the following stability variation: How much does a convex body *K* deviate from a convex body of constant width if the perimeter of the projections of *K* is approximately constant? The 3-dimensional case was studied by Campi [208], and Bourgain and Lindenstrauss [170] solved the related stability question regarding areas of projections. The paper [441] contains far-reaching generalizations of Minkowski’s theorem, involving the stability of the mean width functional in the following way. The width function of a convex body *K* in \({\mathbb E}^n\) is nearly constant if the mean width function of the projections of *K* onto hyperplanes is nearly constant. And if the latter function is assumed to be nearly constant, then there exists a set of constant width near *K* in the Hausdorff sense. Groemer [463] introduced and studied measures for a so-called spherical deviation of a convex body from convex bodies of a special class, like that of constant width. Introducing also measures for normals of convex bodies, stability results are proved, i.e., inequalities which show that convex bodies cannot deviate too much from the class of constant width bodies if their corresponding deviations of the normals are small.

Pommerenke [945] derived estimates on the capacity (or the transfinite diameter) of planar compact sets in terms of geometric quantities of them, such as width, area etc. within this framework, he investigated also curves of constant width.

Positive centers of convex curves in the plane were introduced by Gage to sharpen Bonnesen’s isoperimetric inequality (references are given in [562]). The authors of [562] investigated positive center sets, and they also gave a characterization of them for given curves of constant width based on the notion of inner parallel bodies; this work is continued in [911], again referring to constant width sets.

For a closed convex surface

*S* in

\({\mathbb E}^3\), let

*d*(

*K*) denote the least upper bound of the intrinsic distances between point pairs in

*S*, i.e., the so-called geodesic diameter of

*S*. Relating this to the usual diameter

*D*(

*S*) of

*S*, Makuha (see [763] and [764]) proved the inequality

$$ d(S) \le \pi D(S)/2\,, $$

with equality iff

*S* is the boundary of a body of revolution having constant width. The analogous statements hold in

\({\mathbb E}^n\), as confirmed in [762]. The 3-dimensional case was independently reproved by Zalgaller [1209], where also the question for the respective lower bound on

*d*(

*S*) for

*S* having constant width is posed.

Let there be given two parallel lines in Euclidean space which have at most distance *d*. In [377], it was shown that the lengths of the segments intercepted by a body *C* of constant width *h* on such lines differ at most by \(2(2dh)^{\frac{1}{2}}\), not depending on the shape of *C*.

In [221] inequalities for so-called relative quantities (suitably defined by two planar given figures, one containing the other) are derived, and an interesting characterization of sets of constant width is obtained which is related to the notion of relative diameter.