In these notes we mainly collect results on notions related to that of constant width via the common headline *cross-section measures*; the notions of *width* and *maximal chord length* are special cross-section measures, and we try to reflect the literature on further such concepts yielding natural analogues to constant width sets within the framework of projection and section functions.

**Cross-Section Measures and Associated Bodies**

Considering an *n*-dimensional convex body *K*, one can define a couple of natural projection and section functions on the unit sphere which are related to the concept of constant width as somehow analogous notions, like, e.g., constant brightness, constant section, or constant *HA*-measurement. More precisely, already in § 7 of the monograph [160] the so-called outer and inner cross-section measures of convex bodies are introduced, the outer ones meaning projection measures, and the inner ones section measures. Thus, for the following notions we refer to § 7 and § 15 of [160], to Gardner’s monograph [401] (mainly to Chapters 3, 4, 6, and 8 there) and to the survey [783]; also our Chapter 12 is related. For a convex body *K* and any direction \(u \in S^{n-1}\), the \((n-1)\)-volume of the orthogonal projection \(K|u^\perp \) of *K* onto the \((n-1)\)-dimensional subspace normal to *u* is called the *outer (n−1)-dimensional cross-section measure* or *brightness* \(V_{n-1}(K|u^\perp )\) of *K* with respect to *u*. It turns out that \(V_{n-1}(K|u^\perp )\) is the restriction of the support function of the *projection body* of *K* onto \(S^{n-1}\). Any projection body is a *zonoid* centered at the origin, and if *K* is a polytope *P*, then its projection body is even a *zonotope*, i.e., a finite vector sum of line segments parallel to the facet normals of *P* (zonoids are the limits of zonotopes in the Hausdorff metric; see Chapter 4 of [401] and also Section 2.11). Any zonoid with the origin as midpoint is the projection body of a certain class of convex bodies, and the question how much information about this class and its representatives we can get from the given projection body is a typical “tomographic” problem. On the other hand, we write \(\max V_{n-1}(K \cap (u^\perp +tu))\) for the *inner (n−1)-dimensional cross-section measure*or *HA-measurement* (or maximal section function) of *K* with respect to \(u \in S^{n-1}\). Here we have that \(\max V_{n-1}(K \cap (u^\perp +tu))\) is the restriction to \(S^{n-1}\) of the radial function of the *cross-section body* *CK* of *K* (cf. § 8.3 of [401]). We replace now these \((n-1)\)-dimensional measures by one-dimensional ones, referring to 1-subspaces \(s_u\) or lines \(l_u\) in direction *u*. In the projection case we get the *outer 1-dimensional cross-section measure* \(V_1(K|s_u)\), better known as *width* of *K* in direction *u*, and in the section case the *inner 1-dimensional cross-section measure* \(\max V_1(K \cap l_u)\), where \(l_u\) runs over all lines parallel to \(s_u\). Thus, \(\max V_1(K \cap l_u)\) describes the *maximal chord length* in direction *u*. It turns out that \(V_1(K|s_u)\) and max\(V_1(K \cap (l_u)\), \(u \in S^{n-1}\), describe the restriction of the support function and of the radial function to \(S^{n-1}\) of the same convex body corresponding with *K*, namely the *difference body* \(DK = K + (-K)\) of *K*. In addition, the closely related concepts of *intersection bodies* (see Chapter 8 of [401]) and *chordal symmetrals* (cf. Chapter 5 of [401]) should be mentioned here. The radial function of the intersection body *IK* of *K* is given by \(V_{n-1}(K \cap u^\perp )\), and the radial function of the chordal symmetral of *K* by \(V_1(K \cap s_u)\). It is obvious that the questions how far *K* is determined, if only such projection and section functions of it are known, naturally belong to the field of geometric tomography, basically described in [401]. In other words: given such a projection or section function (or combinations of them) on the unit sphere, how much is known about the class of original convex bodies creating these functions? For example, if we additionally assume central symmetry of the bodies of such a class, then the uniquely determined representatives of such classes are called *Blaschke body* (for \(V_{n-1}(K|u^\perp \))), *central symmetral* (for \(V_1(K|s_u)\) as well as max\(V_1(K \cap l_u)\)), equal to \(\frac{1}{2} DK\), and *chordal symmetral* (for \(V_1(K \cap s_u)\)). Whereas *K* is mostly assumed to be a convex body, its cross-section body, its intersection body, and its chordal symmetral are, in general, no longer convex, but still starshaped. (Due to this, many authors also study such associated bodies for given starshaped sets, for instance referring to intersection bodies.) Of course, all these considerations can be analogously extended to measures of projections onto or sections with affine flats of intermediate dimensions *k*, with \(2 \le k \le n-2\) and \(n \ge 4\). For example, this yields then the notions of *bodies of constant outer and inner k-measures*, see, e.g., [359], and cf. also Subsection 13.3.3. For example, the following statement was first formulated by Blaschke and Hessenberg [136]: For \(2 \le k \le n-1\), let all projections of a convex body *K* in \(\mathbb {E}^n\) onto all *k*-dimensional subspaces of \(\mathbb {E}^n\) be of constant width in these flats. Then *K* is itself of constant width. Montejano [848] sharpened this result by showing that the condition \(k > n/2\) is sufficient. Comprehensive discussions of this framework are given in § 7 and § 9 of [160], Chapters 3, 4, 6, 7, 8, and 9 of [401], [442], [443], and also in the survey [783]; many more recent results are widespread in the literature and partially presented below. In fact, due to the theme of our book, only the following restricting assumption is relevant: that the functions introduced above are constant over \(S^{n-1}\), or analogously constant regarding intermediate dimensions, then using Grassmannians. This constancy can be assumed for only one measure (yielding, e.g., bodies of constant brightness), or for more than one measure referring to the same body (yielding, e.g., bodies which are at the same time of constant width and of constant brightness). From the arguments above it follows that any body of constant width is equivalently a body of constant maximal chord length and has a ball as difference body (or central symmetral), and any body of constant brightness has a spherical projection body, and so on. For more characteristic properties which are closely related we refer also to our Chapter 3.

**Constant Brightness**

We start with *bodies of constant brightness*, i.e., with those convex bodies in \(\mathbb {E}^n\) whose projection bodies are balls, see also Subsection 13.3.2. For more background we refer to [272, A 10], [401, Chapters 4 and 9], and [783]. Inspired by the dissertation [534] of Herglotz, Blaschke (see part IV of the “Anhang” from [132]) gave a suggestive example of a 3-dimensional body of constant brightness, and he derived an exact criterion for a convex body to be of constant brightness, which is also discussed in Section 68 of [160]. Namely, a convex body with suitable curvature properties is of constant brightness iff the sum of the reciprocal Gaussian curvatures in two points in parallel support hyperplanes is constant. The following property, extending a result of Berwald [102], is due to Schneider [1033]: a convex body in \(\mathbb {E}^n\) has constant brightness iff the integral of the surface areas of its sections by hyperplanes orthogonal to a direction *u* is not depending on *u*. Further related problems and results were obtained in [357], [358], [360], [632], [468], and [209]. In particular, a centrally symmetric convex body of constant brightness is a ball, see [358]. Firey (see [357] and [358]) developed techniques to construct nonspherical bodies of constant brightness with certain smoothness assumptions, based on analogous ideas of Vincensini (cf. [1148], [1149], [1150], [1151], and [1152]) referring to Minkowski addition. Vincensini investigated the classes of convex bodies having a prescribed difference body by a method of extending linear (in the sense of vector addition) families of convex bodies. The subcase of spherical difference bodies clearly yields families of bodies of constant width. Later developments of this technique are presented in [1153] and [1154]. Inspired by this method, Firey developed in [357] and [358] the analogous methods for bodies of constant brightness, replacing Minkowski addition by the so-called Blaschke addition of two convex bodies (namely, using the sum of their surface area functions). For example, the Blaschke sum of two constant brightness bodies has constant brightness. Klee [632] posed the problem of constructing for each *k* between 1 and \(n-1\) a nonspherical convex body of constant outer *k*-measure. Firey constructed in [359] examples (even bodies of revolution) for any such *k*, reobtaining for \(k=2\) and \(n=3\) also Blaschke’s example from [132] mentioned above, and generalizing it directly to higher dimensions (yielding the class of so-called Blaschke-Firey bodies). Gronchi [468] proved that within the family of convex bodies of revolution in \(\mathbb {E}^n\) having constant brightness, the Blaschke-Firey bodies are the unique (up to congruence) members of minimum volume. See Theorem 14.2.3. This confirmed a conjecture from the older paper [209], where also a necessary condition for volume-minimizers of fixed constant brightness in \(\mathbb {E}^3\) in terms of surface area functions is derived. This yields a plausible conjecture on volume-minimizers of fixed constant brightness, namely that their surface area measure is supported by, and uniformly distributed in, four alternate spherical triangles in the partition of the unit sphere by three mutually orthogonal great circles. Also related to volume-minimizing conditions is [356]. Blaschke’s exact condition for bodies of constant brightness mentioned above was extended in [359] to sufficiently smooth bodies of constant outer *k*-measure in \(\mathbb {E}^n\), see also Section 13.3.3. In [931] an inequality is proved, whose case of equality characterizes the affine images of bodies of constant brightness. The papers [249] and [254] refer to shadow boundaries, integral formulae, and principal radii of curvature for ovaloids (i.e., compact surfaces in 3-space having continuous positive Gaussian curvature) of constant brightness in \(\mathbb {E}^3\). For proving an extension of a result of Aleksandrov (see [12] and our subsection on combinations below), Chakerian and Lutwak [240] introduced a generalization of the notion of constant brightness.

Another generalization, the so-called *constant joint brightness* of a family of convex bodies, was introduced in [725] and obtained as extremal case in Aleksandrov-Fenchel type inequalities referring to mixed brightness integrals.

**Constant** *HA* **-Measurement**

The function max\(V_{n-1}(K \cap (u^\perp + tu))\) is called *HA*-measurement of the convex body *K* since this inner cross-section measure has physical applications in the study of Fermi surfaces of metals (determining the surfaces from its *HA*-measurements which themselves can be found by the de Haas-von Alphen effect (see, e.g., Shoenberg [1062] and Mackintosh [746]). For a convex body *K* in \(\mathbb {E}^n\) containing the origin, Klee [632] asked whether its *HA*-measurement is only constant when *K* is a ball (Klee asked even more about this section function, see A11 in [272] and below). This assumed constancy means that the cross-section body *CK* is a ball (see also § 8.5 of [401]). For origin-symmetric bodies this is the case (cf. [385], [726], and [332]), but the authors of [891] proved that, in dimension 4, there are convex bodies (of revolution) that are not balls but for which max\(V_{n-1}(K \cap (u^\perp + tu))\) is constant. They announced the same result for all dimensions. In [892] the same authors showed that the answer to Klee’s question is negative in all dimensions \(n > 2\), even for bodies of revolution. The authors of [403] constructed a centrally symmetric convex body *L* and a nonsymmetric body *K* having the same *HA*-measurement function. The situation for dimension 2 is trivially the same since there are bodies of constant width that are not circular. We mention here the much older paper [1211], where such questions were already investigated. Related results on sections with and projections onto lower dimensional subspaces are derived in [991]. However, if both functions \(V_{n-1}(K|u^\perp )\) and max\(V_{n-1}(K \cap (u^\perp + tu))\) are constant for the same body *K*, then *K* is a ball centered at the origin, see [755].

**Constant Section**

We switch now to classes of convex bodies or starshaped sets which (should be better called *of constant subspace section*, but) are called *of constant section*. If we formulate “sets/bodies of constant \((n-1)\)-section”, then we mean those sets whose intersection body is a ball. Intersection bodies were introduced by Lutwak [743] for getting reasonable dualizations of basic notions of convexity (like mixed volumes). Meanwhile this notion is established and important in many subfields of convexity, like in the dual Brunn-Minkowski theory and in the field of valuations. And it turned out that they also helped to solve famous problems like the Busemann-Petty problem, see, e.g., [399], [400], [402], [649], and [650] (see Section 2.11). Intersection bodies are also comprehensively discussed in Chapter 8 of [401] and in Chapter 5 of [1124], in the latter reference due to their importance for the isoperimetric problem in Minkowski spaces based on Busemann’s definition of area. In [405] the injectivity of the spherical Radon transform was used to prove results on bodies of constant *k*-section (for *k* between 1 and \(n-1\)), thus related also to the notion of chordal symmetral and its generalizations. It is shown that if *L* is a centrally symmetric star body for which all *k*-dimensional sections through the center have the same *k*-dimensional volume, then *L* has to be a ball. On the other hand, nonspherical convex bodies are constructed which contain the origin and all whose *k*-sections through the origin cut out the same *k*-volume. If *L* is a star body (not centrally symmetric) with *k*-sections of constant volume for two different values of *k*, where the *k*-sections all pass through some interior point, then *L* is necessarily a ball centered at this point. In [404] these investigations are continued. In some sense dual to Aleksandrov’s result on *k*-projection functions of centrally symmetric convex bodies, Funk [385] confirmed the same for *k*-section functions. The authors of [404] show that no set star-shaped (with continuous radial function) with respect to the origin is determined, up to reflection in the origin, by its *k*-section function, in contrast to the situation for convex bodies and their projection functions. Furthermore, the family of such star-shaped sets that are, up to reflection in the origin, determined by their *k*-section functions for all \(k=1,2,\dots , n-1\), is nowhere dense in the set of all such star-shaped sets. The paper [404] contains also interesting results about chordal symmetrals which are naturally related to point *X*-rays (see Chapter 5 of [401]). And in [755] it was shown that if the chordal symmetral and the difference body of a convex body *K* are centered balls, then *K* itself is a centered ball. Chordal symmetrals and their generalizations, called *i*-chordal symmetrals, are widely discussed in Chapters 5 and 6 of the monograph [401]. In particular, we mention here the famous equichordal problem (see § 6.3 in [401], where it is suitably discussed for star bodies). Namely, an *equichordal point* of a Jordan curve *C* is a point inside the curve such that every chord of *C* through this point has the same length. The equichordal problem (see also the notes of our Chapter 11, below the headline "Affine geometry") asks whether there is a curve with two distinct equichordal points. This problem has a rich and long history. However, since it is not in the direct focus of our aims here, it should suffice to cite only [631], the discussion A1 in [272], § 6.3 in [401], where also generalized equichordal problems are presented, and the two papers [993] and [994]. The first one contains a complete solution of the equichordal problem, and the second paper is a useful, critical discussion of contributions, nicely written for a wide readership. The answer to the equichordal problem is “no”, i.e., for one given set two such points cannot exist. However, on the sphere curves with two equichordal points exist, see [1084], and Petty and Crotty [933] showed that there are Minkowski spaces in which convex bodies with the same property exist (the latter paper contains also related results on spherical neighborhoods and constant width sets).

Nevertheless, equichordal bodies *K* (i.e., bodies all whose chords passing through some \(p \in K\) have lengths 1 or, equivalently, whose chordal symmetrals are balls) yield also interesting research problems. Groemer (see [459] and Section 6 of [460]) investigates the stability of inequalities referring to quantities of them, e.g., bounding their volume. In [215] various geometric properties of equichordal curves are derived, and also relations to the notion of transnormality are clarified. And in [217] the notion of diametrical submanifold of a Euclidean space with respect to a center *p* is defined, which is a special case of tangential symmetry in convexity. Namely, such a submanifold admits a tangent-preserving diffeomorphism such that the chords connecting the points on the submanifold with their images pass through *p*. For compact, connected submanifolds satisfying a certain condition the property of being diametrical is the same as being centrally symmetric and, if that is the case, there is just one center. In the present paper, the authors define equichordality for the submanifolds and study the case when a submanifold admits an equichordal and a diametrical diffeomorphism with respect to the same center. For a certain modification of the equichordality notion, in [214] a characterization of spheres is proved. And in [260], so-called *g*-chordal sets as a generalization of equichordal sets are investigated, yielding an interesting Crofton-type formula and also a lower bound of the area enclosed by such *g*-chordal curves. For similar results we refer also to [248]. Eggleston considered in [312] also polars of constant width sets. If a body of constant width has the origin *o* as interior point, then its polar has an *equireciprocal point*, that is, the sum of the reciprocals of distances from *o* to *p* and from *o* to *q* is the same for all chords *pq* of the polar body passing through *o*. If *S* denotes the surface obtained from the boundary of this polar body via inversion at the unit sphere \(S^{n-1}\), then *o* is an equichordal point of *S*. In the planar case, the analogue of *S* is the pedal curve of the constant width set which was the starting point here (see also [617]). For more about sets with equireciprocal points we refer to [333] and Section 2 of [635], and inequalities for convex sets with one such special point (like an equichordal or an equireciprocal one) are derived in [413] and [460].

Weakly related to the notion of intersection body, Stephen asked in [1099] whether there is a convex body in \(\mathbb {E}^n, n > 2,\) which is not centrally symmetric and whose *convex* intersection body (see [1099] for a definition) is a Euclidean ball.

**Constant Girth**

Not directly connected with our framework of cross-section measures, but nevertheless closely related are the so-called bodies of *constant girth* and of *constant k-girth*. This topic has to do also with Minkowski’s projection theorem discussed already in our Section 13.1. A more general and broader discussion is given in § 3.3 of Gardner’s book [401]. If for \(1 \le k \le n-1\) and a convex body *K* in \(\mathbb {E}^n\) the value \(V_k(K|u^\perp )\) is constant for all directions *u*, where \(V_k\) denotes the *k*-th intrinsic volume of \(K|u^\perp \), then *K* is said to be of *constant k-girth* (this definition is due to Chakerian [231], yielding for \(k = n-2\) the usual notion of *constant girth*). It was shown in [231] that under certain smoothness assumptions every body of constant outer *k*-measure must have constant *k*-girth, and Firey [359] could remove these boundary restrictions. The converse seems to be open, although Firey [359] confirmed it for bodies of revolution. And Theorem 3.3.13 in [401] says that a convex body in \(\mathbb {E}^n\) is of constant width iff it is of constant 1-girth. Extending a result of Aleksandrov from [12], Chakerian and Lutwak use in [240] also a generalization of the notion of constant girth. And in [744] bodies of constant girth occur as extremal cases in inequalities referring to quermassintegrals of mixed projection bodies. It is natural to ask the following: does a convex body in 3-space have almost constant width if the perimeter of its projections is nearly constant? Goodey and Groemer [441] treat this stability problem associated with Minkowski’s projection theorem and extend their results also to the *n*-dimensional situation.

**Combinations**

To combine different projection and/or section functions, we have the possibility to stay only with one type (e.g., only with projection functions), or to combine section functions with projection functions. For both cases, the famous Nakajima problem (see [887]) is a good starting point: Are the balls the only convex bodies in \(\mathbb {E}^3\) being of constant width and of constant brightness? For related observations and problems we refer also to the paper [233]. For convex bodies with twice continuously differentiable boundary, this was confirmed by Matsumura (see § 15 in [160], page 82 in [238], A10 in [272], and Theorem 3.3.20 in [401]; Chakerian [231] proved an analogue for three-dimensional Minkowski spaces, and recently Stepanov [1098] gave an alternative proof of the original statement). These boundary conditions could be deleted by Howard [557]; he confirmed this nice ball characterization in \(\mathbb {E}^3\) in full generality and gave also extensions of this result. Continuing this, the authors of [559] proved the following interesting result: let *K* and *L* be convex bodies in \(\mathbb {E}^n\), and let *L* be centrally symmetric and satisfy a weak regularity and curvature condition. Assume that *K* and *L* have proportional first and *k*-th projection functions, where \(2 \le k < (n+1)/2\) or \(k=2, n=5\). Then *K* and *L* are homothetic. Assuming that *L* is a Euclidean ball, one thus obtains characterizations of Euclidean balls as convex bodies of constant width and constant brightness. Another strong generalization in this direction was proved in [558]: in \(\mathbb {E}^n\), let *L* be a centrally symmetric convex body having \(C^2_+\) boundary, and *K* be a convex body with \(C^2\) support function. Excluding the two exceptional cases \((i, j)=(1,n-1)\) and \((i, j)=(n-2,n-1)\), it is shown that *K* and *L* are homothetic if their *i*-th and *j*-th projection functions are proportional. The author of [500] investigated whether a convex body of class \(C^2_+\) in \(\mathbb {E}^n\) with constant *i*-brightness and constant *j*-brightness is a ball for \(i < j\). Unfortunately, his approach contains gaps. Hug [564] proved, even more general, that if *K* is a convex body in \(\mathbb {E}^n\) and *i*, *j* are two integers with \(1 \le i < j \le n-2\) such that \((i, j) \ne (1,n-2)\), then the constancy of the *i*-th and *j*-th projection function implies that *K* is a ball. This is only a partial result from [564], more general ones on homothety classes of convex bodies are derived there. Regarding homothety classes of convex bodies and projection functions we also refer to § 3.1 of [401]. Related to these results, in [442] and [443] (the first paper being a survey) it is investigated to what extent convex bodies are determined by their projection functions, also with respect to intermediate dimensions *i*. If *K*, *L* are centrally symmetric convex bodies and at least one has dimension larger than *i*, then already Aleksandrov [12] showed that equality of both *i*-dimensional projection functions implies that *L* is a translate of *K*. In [442] and [443] such investigations are continued. For general convex bodies, clearly the *i*-dimensional projection volume cannot distinguish between *K* and any translate or reflection of it. First, most (in the sense of Baire category) convex bodies are determined by any single *i*-projection function, with *i* going from 2 to \(n-1\). Second, there is a dense family of convex bodies whose members are not determined by knowing all the *i*-th projection functions with *i* going from 1 to \(n-1\). And it is proved that, in the usual Baire category sense, most convex bodies are determined, up to translation or reflection, by the combination of their widths and brightnesses in all directions. Obviously, in all these considerations the subcases referring to bodies of *constant* *i*-*measures* are directly interesting for our purpose here.

Bringing also section functions into the game, we mention once more: if \(V_{n-1}(K|u^\perp )\) and max\(V_{n-1}(K \cap (u^\perp + tu))\) are both constant for all directions *u*, then *K* is a ball centered at the origin, see [755]. The one-dimensional analogue says that if the chordal symmetral and the difference body of a convex body *K* are centered balls, then K is a centered ball (see again [755]). Perhaps the little field of combining suitably projection and section functions still hides many interesting questions. And we remark that also some of the problems posed at the end of Chapter 8 of Gardner’s book [401] are closely related to the philosophy of this subsection here.

**Methods**

From the viewpoint of methods used for studying projection and section functions in the way described above, especially the books [461], [1039], and [401] are fundamental. In 1998, Koldobsky [649] found a general Fourier transform formula which could be used to provide a unified approach to several geometric problems of the type discussed here. In this booklet, he recalls the basic methods of Fourier analysis and convex geometry, and some facts concerning Radon transforms and spherical harmonics, and then applies them to the relevant geometric problems, e.g., related to volumes of sections of convex bodies. We refer also to [650] with similar aims. A related work from 2014 is [992], similarly explaining analytical methods needed for the study of projection and section functions in convexity, also with many exercises. More general, we refer here also to various surveys collected in Part 4 (Analytical aspects) of the Handbook [483].