Bodies of Constant Width in Minkowski Spaces

Abstract

In Euclidean space, the length of a segment depends only on its magnitude, never on its direction. However, for certain geometrical problems the need arises to give a different definition for the length of a segment that depends on both the magnitude and the direction.

Some mathematician, I believe, has said that true pleasure lies not in the discovery of truth, but in the search for it.

Leo Tolstoy

Appendices

Notes

The origins and basic definitions of Minkowski geometry are connected with names like Riemann, Minkowski, Banach, and Busemann. The field of Minkowski geometry can be located at the intersection of Finsler geometry, Banach space theory, and convex geometry, but it is also closely related to many other fields. An example is distance geometry (in the spirit of Menger and Blumenthal [141]), another one is combinatorial geometry (see, e.g., [151, Chapters II and V]). Minkowski geometry was also enriched by many results from applied disciplines such as operations research, optimization, theoretical computer science and location theory. For an excellent book covering mainly the analytic part of the theory see [1124], and for surveys covering hundreds of papers in Minkowski geometry widespread in very different fields one should consult [795] and [793]. Since it is very natural to carry over the concept of constant width to (finite dimensional) real Banach spaces, the number of publications in this little field is large. We try to give an overview which is as complete as possible. In former expositions, many of these publications are also cited, in particular in the three surveys [238], [527], and [793]. Many new results were published later. Summarizing, many contributions (also some older ones) cannot be found in the mentioned three surveys. Thus, we aim to give here an overview with update which is as complete as possible. Of course, we use the introduced notation as far as possible. Since in these notes here we always refer to Minkowski spaces (and also to more general Banach spaces), we omit the addendum $\mathcal {G}$ everywhere and we do not extra mention this. Note that sometimes in the literature the notion “constant relative width” occurs, meaning the same concept (but to avoid confusion, we should mention here also the related, but different concept of relative differential geometry; see the survey [80]). And when using the word orthogonality, we mean (if not otherwise said) the most common orthogonality concept for normed spaces, namely that of Birkhoff orthogonality, see [24]. This yields also that, when speaking about double normals, we naturally replace diameters, important for the Euclidean case, by affine diameters (see the excellent survey [1076] on this notion).

Basic Geometric Properties of Constant Width Bodies

The basic theorems from Section 3.1 in this book here and Section 2 in [238] about the ball shape of the difference body of constant width sets, the existence and length of diametral chords, their orthogonality to supporting strips, their coincidence in the parallel case as well as their double normal property (here for smooth and strictly convex balls) are derived in [316] and [238, Section 2], see also [336] and [509] and the comprehensive representation in [793, § 2.1]. For example, we have the following basic statements: A convex body in n-dimensional Minkowski space is of constant width h iff its difference body coincides with the unit ball, and iff all its diametral chords have length h. Thus, the only centrally symmetric constant width bodies are also in Minkowski spaces the balls; e.g., [58]. And if the unit ball is smooth and strictly convex, then a convex body is of constant width iff any two parallel normals of it coincide. Petty and Crotty [933] proved that a body of constant width having an equichordal point is a ball of the norm (an alternative approach to an important partial step in the proof was established in [509], see also [380] for $n = 2$ and $n = 3$). It was also shown in [933] that there are Minkowski spaces with convex bodies having exactly two equichordal points. In the mentioned paper [509] results on diametral chords of constant width sets, also derived in the papers [506], [511], [512], [508], and [1072], are extended to normed planes. Vrećica [1159] confirmed that a convex body is of constant width iff for all points a, b from its interior, there is a set of constant width contained in this interior having a, b itself as boundary points. An important theorem for normed spaces is the monotonicity lemma, see subsection 3.5 of the basic expository paper [795] or [1124, Lemma 4.1.2]. Heppes [531] proved a theorem which can be interpreted as monotonicity lemma for sets of constant width in the Euclidean plane. One of the implications was extended to strictly convex normed planes by Grünbaum and Kelly [490], and in [59] the result from [531] was completely extended: Any hyperplane section S of an n-dimensional convex body of constant width divides that body into two compact, convex sets such that at least one of them has the same diameter as S. For $n=2$ this even characterizes the sets of constant width. We refer also to the related papers [694] and [54]. In the latter paper it is shown that in higher dimensions at least the following holds: If the convex body is partitioned by a continuous family of hyperplane sections $S(t), 0 \le t \le 1$, then the diameter of S(t) is an unimodal function of t. Closely related is also [57]: In a strictly convex, smooth and n-dimensional Minkowski space a convex body with the same boundary restrictions is said to have property (P) if any manifold $M_0$ homeomorphic to the ($n-2$)-dimensional sphere and lying on the boundary of the body splits its boundary into two compact manifolds, one of them having the same Minkowskian diameter as $M_0$. It is shown that if a convex body has constant width, then (P) holds for it. Vice versa, if the body satisfies (P) and has at least two diametral chords, then it is of constant width. And still related is the paper [52]: a convex body in an arbitrary normed plane is of constant width iff it is splitted by every chord I of it into two compact convex sets $K_1$ and $K_2$ such that I is a Minkowskian double normal of $K_1$ or $K_2$. In [55] the following result, which was announced by Hammer and Smith [510], is established for normed planes: If all binormal chords of a plane constant width set bisect its Minkowskian perimeter, then it is a ball. This paper contains also interesting results on normed planes whose unit circles are equiframed curves (i.e., centrally symmetric closed convex curves touched at each of their points by some circumscribed parallelogram of smallest area). Chakerian [231] (see also [336]) established the following well-known theorem for Minkowski spaces: The inradius r and the circumradius R of a convex body of constant width h satisfy $r + R = h$, and the corresponding in- and circumsphere are concentric. Sallee [1001] showed that this result still holds if “constant width” is replaced by “spherical intersection property”, related also to the notion of Minkowskian pairs of constant width. Barbier’s theorem says that any planar set of constant width h has the same perimeter $h \pi$, and it holds analogously in normed planes. Most likely, the first proof of it was presented in [929], and further proofs and variants can be found in the papers [827], [785], [625], [270], and [697]. The latter paper contains even a generalization of Barbier’s theorem, and there it is also proved that for every $\varepsilon > 0$ and every convex body of constant width in a normed plane there exists a convex body of the same constant width whose boundary consists only of arcs of circles in the sense of the norm such that the Hausdorff distance between the two bodies is at most $\varepsilon$. As an approximation result, this generalizes the Euclidean case proved by Blaschke. Let M denote a normed plane with isoperimetrix I, and $M'$ be the Minkowski plane with I as unit circle. In [625] nice results on curves of constant width in such pairings of normed planes are established. For example, if C has constant width h with respect to $M'$, then $A(C) + A(C,-C) = h \frac{A(I)}{2}$ holds, where A denotes area and $A(C,-C)$ the mixed area of C and $-C$. And if the perimeter L(C) of such C is measured in M, then $L(C) = h A(I)$. As already mentioned an analogue of Barbier’s theorem is derived, too. The paper [223] is based on the same concept with $M'$ and I. It is shown there that if C is a smooth curve of constant width in $M'$ each of whose diametral chords bisects the $M'$-perimeter (or the area), then C is homothetic to I. In [22], examples are provided which show that if one uses Birkhoff orthogonality, then the Makai–Martini characterization of curves of constant width (Theorem 4.4.1) cannot be extended to normed planes. The authors present also further results on intersecting orthogonal chords of unit circles of Minkowski planes, based on Birkhoff and also on James orthogonality. On the other hand, in [25] the new notion of affine orthogonality is introduced, and for this type of orthogonality the Makai–Martini characterization has some extension. The paper [25] contains also other characterizations of sets of constant width, e.g. via a double-normals property.

Special Bodies of Constant Width

There are different possibilities to define the Minkowskian analogue of the Reuleaux triangle. Analogous to the Euclidean situation, Ohmann [898], Chakerian [230] and Wernicke [1197] defined it as intersection of three circles, each of radius h and centered at one of the vertices of an equilateral triangle, and they described it as extremal figure regarding certain metrical problems (we refer to our subsection below on inequalities). Wernicke [1197] proved that the planes with parallelograms or affine regular hexagons as unit circles are the only ones in which Reuleaux triangles exist as Minkowski circles; this was reproved in [785]. Constructions of constant width figures and, in particular, Reuleaux polygons are presented in Chapter 4 of the monograph [1124], [672], and [667]. Reuleaux polygons are also investigated in [509] and [997]; Hammer [509] obtained analytical representations of special constant width curves, giving a specific formula (involving suitably properties of the unit ball) always leading to a curve of constant width. Petty [929] used the Minkowskian analogue of “evolute” to construct special curves of constant width. The authors of [270] defined notions like Minkowskian curvature, evolutes, and involutes for polygons of constant width. They confirmed that many properties analogous to those of the smooth case, which was studied before in [269], are preserved under corresponding iterative procedures. For example, the iteration of involutes of polygons of constant width generates a pair of sequences of polygons of constant width with respect to the given norm and its dual, respectively. It turns out that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the starting polygon. Further geometric properties of Minkowskian Reuleaux triangles are discussed in [785], [997] and [965].

Completeness and Completions

We recall that a bounded subset C of a Minkowski space is called complete or diametrically complete if it cannot be enlarged without increasing its diameter, and any complete set C having the same diameter as a convex body K with $K \subset C$ is said to be a completion of K. Important early references on complete sets in Minkowski spaces are [316], [160], [457], see also [151, Chapter V]. We will survey now results on complete sets, completion procedures and applications of these notions in Minkowski spaces and in more general Banach spaces. For normed planes, the notions of completeness and constant width are equivalent, and in higher dimensions constant width still implies completeness, see [336, § 2] and [316] for an explicit proof, and also Theorem 10.3.3 and Corollary 10.3.1. The converse implication was confirmed for $n = 3$ by Meissner [817] and for higher dimensions by Kelly [616], but their proofs were erroneous. Eggleston [316] constructed an explicit example showing that for $n > 2$ the converse is no longer true, even for norms that are strictly convex and smooth, and he also constructed an example with polyhedral unit ball. The characterization of those Minkowski or Banach spaces in which completeness and constant width are equivalent properties became a famous research topic, see below and, e.g., [861] and [788]. The norms with this property are called perfect norms. This name is due to Karasev [606], and still no characterization of this important class of norms is known. Minkowski spaces with perfect norms are precisely those in which the family of complete sets is convex, or the family of completions of any given set is convex. Karasev also showed that if a norm is perfect and strictly convex, then the intersection of the unit ball with any translate of it is a summand of it. Polovinkin [940] defined an interesting notion which is closely related to perfect norms. He rediscovered an elegant completion procedure presented originally by Maehara [747] for the Euclidean case. This concept is also discussed in subsection “Related concepts” below, in these notes. According to Sallee [1002] and Polovinkin [940], this way of completion even works for a class of Minkowski spaces, namely those whose unit ball is a so-called generating set. A convex body is called a generating set if any intersection of translates of it is also a summand of it. Maehara [747] was the first who proved that Euclidean balls are generating sets, of course without using this name, and Polovinkin [940] introduced and extended this notion. The topological and algebraic properties of generating sets, the properties of related notions like M-strongly convex hulls and support functions of M-strongly convex sets (see below) were deeply studied in [64]. From results derived in [747] and [1002], Polovinkin could conclude that if the unit ball of a Minkowski space is a generating set, then the norm is perfect. (Note that at the end of this subsection we discuss types of generating sets which are known until now.) Based on [1002] and [64] it was also shown by Karasev [606] that in a reflexive Banach space the so-called Maehara set of a given compact convex set is of constant width. Since Maehara’s type of completion is not extendable to all Minkowski spaces, Moreno and Schneider [864] were motivated to look for further completion procedures extendable to all Minkowski spaces. Based on results of Bückner [192] (see also [966] for a 2-dimensional earlier version, and [677] for finding independently an important step in this approach), they succeeded with a constructive completion procedure in general Minkowski spaces and showed also that their procedure yields a locally Lipschitz continuous selection of the completion mapping. And also the elegant completion procedure of Maehara was studied in [864], also having locally Lipschitz continuous extensions.

Modifying two well-known characteristic properties of bodies of constant width, Moreno and Schneider [862] presented a new characterization of complete sets in Minkowski spaces. More precisely, they proved that a convex body is complete iff the width of each regular supporting slab of it equals its diameter (equivalent to the assertion that the length of each regular diametral chord equals the diameter). Sharp estimates for circum- and inradius of a complete set of given diameter were derived, and it was also verified that in a generic Minkowski space of dimension $> 2$ the family of complete sets is not closed under the operation of adding a ball; also new results about perfect norms were obtained (see also [971]). In [861] properties of the family of all complete sets of diameter 2 in a given Minkowski space were studied. It was shown that this space (obtained when metrized by the Hausdorff distance) is generally not convex, and that it is not necessarily starshaped. A characterization of those spaces in which this set is starshaped was given, via some properties which were already obtained in [862]. Furthermore, it was proved that this space is contractible, and that for polytopal unit balls the space of corresponding translation classes is the union set of a finite polytopal complex. The authors of [863] also studied the mapping $\gamma$ assigning to each compact convex set the family of all its completions and proved that in normed spaces with Jung constant less than 2, $\gamma$ is locally Lipschitz continuous with respect to the Hausdorff metric induced by the respective norm (see also above and [856]). One motivation, interesting for us here, is the fact that $\gamma$ is important for the study of bodies of constant width. It was proved that a Minkowski space has perfect norm if and only if $\gamma$ is convex for every nonempty, compact and convex set (see also [606]). For a compact convex subset C of a Banach space, a closed convex set K containing C with the same diameter is called a tight cover of C. For finite dimensions, there is at least one tight cover of C with maximal volume, and this turns out to be a completion of C and is, for strictly convex norms, even unique (see also again [457]). Groemer [457] discussed also symmetry properties of such completions. The paper [858] is concerned with topological and stochastic properties of the family of all closed convex sets with unique completion. With a sharpened version of a lemma due to Groemer it was shown that, in strictly convex Minkowski spaces, this family is lower porous. This improved a previous result from [457] where this family was shown to be nowhere dense. In contrast to this, there is a stochastic construction procedure which provides a complete set with probability one. This generalizes an earlier result from [81] for the Euclidean plane. If in Minkowski spaces, also higher dimensional ones, a complete convex body is smooth and strictly convex, then it is also of constant width, see [890] and [336, Theorem 3.18]. In [890] also the first example of a norm was found such that the vector sum of a complete set and a ball is not complete. In the latter paper it is also proved that if there exist symmetric points p and $-p$ such that every facet of the unit ball of the space contains one of them, then balls are the only sets of constant width in this space. Also in this paper, sets in Euclidean space are studied which have unique extensions to sets of constant width. Eggleston [316] proved that in any Minkowski space a compact set has the spherical intersection property (SIP) (see below) iff it is complete and iff each boundary point of it has its diameter as distance to at least one other point of it (an alternative proof was given in [336, § 3]). In [313] it was wrongly stated that the only unit balls for which any constant width set is a ball are the parallelotopes. To get the right answer, one has to look for all irreducible sets (cf., e.g., Shephard [1060]) i.e., for all convex bodies K centered at the origin for which the representation $K = Q + (-Q)$ is only possible if Q is centrally symmetric. Yost [1208] showed that for dimension $n > 2$ most (in the Baire category sense) centered convex bodies are irreducible, and that most n-dimensional Minkowski spaces (having a smooth and strictly convex unit ball) have the property that all constant width sets are balls; hence they are complete (see also [1039, § 3.3]). On the other hand, any complete set is a ball iff the unit ball is a parallelotope, see [316] and [1074]. (The 2-dimensional case of the latter statement was already given in [509], and the infinite-dimensional situation was studied in [275] and [374].) Investigating 4-dimensional polyhedral unit balls in view of irreducibility, in [923] examples of Minkowski spaces important regarding the n-ball property and with respect to semi-M-ideals (basic notions from Banach space theory) were constructed. Uniqueness of completion is also important for Borsuk’s partition problem, see [151, § 33] and [788]. In addition, we refer to [77], where investigations from Groemer [457] are summarized and continued, and the relations of completeness and the spherical intersection property in the infinite-dimensional case are investigated. In [179] a one-to-one connection between the generalized Jung constant and the maximal Minkowski asymmetry of the complete bodies in an arbitrary Minkowski space is established. With this starting point, the authors generalize and unify also recent results on complete bodies. The paper [179] contains also results for nonsymmetric unit balls (i.e., for gauges or convex distance functions not necessarily symmetric), and the completeness of simplices plays an essential role for inequalities related to Bohnenblust’s inequality (see [144]) related to the Jung constant. Vrećica [1159] observed that each bounded set has a completion contained in any circumball, which was sharpened in [336]. Sallee [1003] described methods for generating complete sets which contain an arbitrary set, paying special attention to the problem of preassigning boundary points for such sets, and also the preservation of symmetry properties is discussed. Some of these results are reproved in [77], and in addition the extension to infinite dimensions is done there; also Vrećica’s result carries over to infinite dimensions. In [518] all normed spaces are characterized that contain a two-point set with unique completion. They form a strict superfamily of all Banach spaces in which any complete set is a ball, and a strict subfamily of all Banach spaces in which each constant width set is a ball. On the other hand, in finite dimensions there is a completion C satisfying $C \cap S = K \cap S$, where K denotes the bounded set having S as circumsphere, and in infinite dimensions not. In [77] relations between the spherical intersection property and completeness in (finite and) infinite-dimensional Banach spaces are discussed, and it is shown that most, but not all constructions and related results of Sallee, Schulte, and Vrećica also hold for infinite dimensions, where at least one completion always exists, too. In [787] results of the following type are shown: Let $K_1$ and $K_2$ be different complete sets. If there is no inclusion between the sets $K_1$ and $K_2$, then their union cannot be complete; and if the three diameters of $K_1$, $K_2$, and their intersection are equal, then their intersection cannot be complete. On the other hand, a set K and its ball hull have the same completions. In [918] different construction methods are compared, some of which are connected with completions or uniqueness of completions. After a useful survey of known results on this topic, the authors of [920] present new methods to get completions in separable Banach spaces. They use techniques which exist for finite dimensions. The first one is inspired by Theorem 54 in the book [312]. For the second method, the wide and the tight spherical hull are used; the basic idea in finite dimensions is, independently, due to Bavaud [81] and to Lachand–Robert and Oudet [677]. Also in [519] these unions and intersections of completions (i.e., wide and tight spherical hulls) of compact sets are studied. For sets which are not complete, the authors derive boundary properties of their hulls which allow them to characterize diametral points of the original sets, and they also prove that the distance between such a set and the boundary of each of its hulls is always 0, though their intersection may be empty in infinite-dimensional spaces. In [919], several natural relations between balls, complete sets, and completions in Banach spaces are investigated, e.g., conditions for ball-shaped completions. Caspani and Papini [218] discuss various properties concerning complete and constant width sets; in particular, their radii, self radii, and the existence of centers and incenters are studied. Several nice examples, some of them rather pathological, give a fairly complete picture concerning different possible situations. In [219] it was shown that for closed, convex and bounded sets in a Hilbert space constant width and completeness are equivalent notions. If such a set has smooth boundary and for each boundary point there is another one at distance the diameter, then constant width is satisfied. In [767] it was proved that a special Banach space, which is a renorming of $l_2$, contains a complete set without interior points. The paper [768] is among the first to characterize the normal structure of spaces in terms of notions like constant width sets, diametrically complete sets, and sets with constant radius. It starts with a historical survey in that direction. Sets with constant radius and the related family of diametral sets are introduced via the notion of Chebyshev radius. In the previous work [859], Moreno, Papini and Phelps showed that in every normed space a complete set is a set with constant radius. These investigations were continued in [860] by the same authors. There they introduced additionally sets of constant difference and verified the inclusion sequence regarding the four families of sets of constant width, of constant difference, complete ones, and of constant radius. Each of the inclusions can be strict, but under certain conditions (which are also presented) equality holds. For example, the families of sets of constant radius and of complete sets coincide for finite-dimensional spaces; the families of sets of constant width and of constant difference coincide if the Banach space has the Mazur Intersection Property; and all four families coincide in Hilbert spaces. In [797] the authors construct examples showing that in Minkowski spaces a complete set is not necessarily reduced. Continuing such investigations, it was asked in [180] whether a convex body K being complete and reduced with respect to some gauge body (not necessarily centered at the origin) has to be of constant width. This implication is verified for the following large class: the implication holds if K possesses a smooth extreme point. In [917], Papini discussed the role of completeness in approximation theory.

Now we cite results referring to the C(K) spaces, i.e., of all real-valued continuous functions on a compact Hausdorff space K with the maximum norm $\Vert f \Vert =$ max$_{t \in K}\{|f(t)|\}$. We denote by H the family of all closed, convex and bounded subsets of C(K) furnished with the Hausdorff metric. In [855] the notions of completeness and completions were naturally introduced for C(K) spaces, and a characterization of the family of all possible completions of sets from H was given. It was shown that if S is the family of all elements of H having a unique completion, then S is uniformly very porous in H if and only if K is not a singleton. Also following [859] and [860], the author of [856] studied the fundamental properties of the complete hull mapping in C(K) spaces. As above, it associates to each $C \in H$ the set of all its completions. It turns out that this mapping is Lipschitz continuous and has a Lipschitz selection, while it is convex-valued iff K is extremely disconnected. In Euclidean spaces, the mapping is always convex-valued (an example is provided to show that this is not true in a generic finite-dimensional space). In [857] it is shown that in C(K) spaces the following holds: each C within the family of all sets which are intersections of closed balls has a best approximation within the family of completions of C, and within the family of all complete sets. Problems related to vector addition and completion procedures of convex bodies in C(K) spaces are dealt with in [865], and geometric properties of convex bodies in C(K) are shown to characterize the underlying compact Hausdorff space as a Stonean space. It was shown that there is a Maehara-like completion procedure (see [747]), and that the family of all complete sets of diameter h is starshaped with respect to any ball of radius h / 2.

As already announced, we finish this subsection with a little survey on what is known about generating sets and related notions (like results on Minkowski difference if directly related to generating sets, and strong convexity). Note that the system of generating sets is stable under linear transformations and direct sums. 2-dimensional convex bodies are generating (see [406]), like also the unit balls of Hilbert spaces (see [747] and [64]). Generating polytopes were investigated already in [811], together with other interesting polytope classes. In [165] these investigations were continued: the class of all 3-dimensional polyhedral generating sets coincides with the family of strongly monotypic 3-polytopes from [811]. Except for the following class, all these examples are nonsymmetric. Namely, it was shown in [811] that the only centrally symmetric polytopal generating sets are direct sums of polygons and, for odd dimensions, segments. Besides this centrally symmetric case, from [811] the generating n-polytopes ($n > 2$) are only explicitly known if they have at most $n + 3$ facets. In [166] also non-polytopal generating sets are derived. Ivanov [577] derived a criterion for generating sets, yielding a few new (symmetric) examples of generating sets in Hilbert space. After a survey on results of the author, in [941] constructive approaches to completions of sets in reflexive Banach spaces with generating sets as unit balls are presented. Using the machinery of strongly convex analysis (see [939]), the author also established a criterion for unique completion to a constant width body and proposed algorithms for constructing all bodies of constant width containing a given set of the same diameter. Here we also mention [942]. And in [64] the notion of an M-strongly convex set, where M is generating, was introduced. The latter can be represented as the intersection of sets which are translates of the generating set M. The authors described various classes of generating sets and studied properties of corresponding M-strongly convex sets, also strengthening classical Carathéodory-type results and Krein–Milman theorems into this direction. The paper [549] contains interesting separation theorems where generating sets are basically involved. Closely related to the notions discussed here is the study of certain properties of Minkowski sum and difference, which are also important for pairs of constant width, see, e.g., [747], [1002], [167], [439], [849], and [851]. In the latter paper, several topological characterizations of Minkowski summands are given, also yielding a characterization of homothety of convex sets. Also two nice characterizations of the Euclidean n-ball in terms of homological $(n-2)$-spheres or, respectively, in terms of connected components are derived there. We also mention the papers [603] and [604] of Kallay who gave a complete characterization of planar convex bodies which are indecomposable under Minkowski addition within the class of bodies of given width function. In particular, a planar set of constant width 1 is indecomposable with respect to the class of bodies of constant width iff its radius of curvature (which exists almost everywhere on the unit circle) is almost everywhere 0 or 1. There seem to be no analogous results in higher dimensions. Finally, we mention that also in our Sections 7.3 and 7.6 closely related topics are discussed.

Reducedness

It is natural to extend the concept of reducedness also to Minkowski spaces; the greatest part of recent knowledge in this direction is covered by the survey [702]. The first related investigations were done in [700], and the paper [328] contains basic results mainly for the planar case. Again, a convex body in a Minkowski space is said to be reduced if any convex proper subset of it has smaller minimal width, and every convex body contains also a reduced body of the same minimal width. Thus, the following open “dualization” of the isodiametric problem (completely solved for normed spaces by [820], with balls as the unique solution) refers to the much wider family of reduced bodies in normed spaces: Which convex bodies of fixed minimal width have, for any Minkowski space of dimension $n>1$, minimal volume? (For the definition of volume in normed spaces we refer, for example, to [820] and [908].) There are certainly normed planes and spaces for which this can be easily answered, but we are not aware of a systematic investigation in this direction. However, interesting related results nevertheless exist; see, e.g., [53] and [56]. And in [180] it is asked whether a complete, reduced set is necessarily of constant width. This question is even posed for gauges, without the symmetry axiom, and it is answered affirmatively for a large class of norms/gauges. In normed planes the same implication as in ${\mathbb {E}}^2$ holds: each strictly convex reduced body is of constant width (see [328]). Also one might look for norms where the balls are the only reduced bodies. In [700], it was observed that again, for every norm, the only centrally symmetric reduced bodies are the balls (see also [58]). And the following also still holds: Every smooth reduced body is of constant width. Various examples of reduced bodies in normed spaces are collected, discussed, and depicted in the papers [700], [696], [328], [60], and [702]. For fixed minimal width, some of these examples can be “arbitrarily prolonged” like the Euclidean example from [693]. Referring to this, it is conjectured in [702] that in each normed space of dimension $n>2$ there exist reduced bodies of fixed minimal width having arbitrarily large finite diameter. This was confirmed by Richter in [971].

We come now to reduced polytopes in normed spaces, again with an open problem: Do there exist normed spaces of dimension $n>2$ in which no reduced polytopes exist? This question was posed in [702], but in view of [437] (where it is shown that in ${\mathbb {E}}^3$ such polytopes exist) perhaps an affirmative answer becomes less likely. Lassak [696] proved that a convex polytope P in a Minkowski space is reduced if and only if one supporting hyperplane of a minimal-width strip passes through any vertex of it, containing only this vertex of P. Various theorems on reduced polytopes in normed spaces having small vertex numbers were obtained in [60], using new generalized antipodality notions; see also [696] for reduced simplices. It should also be clarified in which normed spaces, for $n>2$, there exist reduced simplices (in the planar situation this is always the case, even with an arbitrary direction of one side of a reduced triangle; see [700]).

In [23], the geometry of reduced triangles is deeply investigated (see also [700]). For example, if the centroid and the incenter of a triangle coincide in the Euclidean plane, this implies equilaterality. In normed planes, in general a wider class of triangles is obtained in this way, but they are still reduced there. On the other hand, equilaterality of triangles in the antinorm is equivalent to their reducedness in the original norm. And equilaterality and reducedness of triangles coincide precisely for Radon norms. Also, reduced triangles can be applied successfully to study the Fermat–Torricelli problem and Steiner minimum trees in normed planes; cf. [237], [792], and [185]. Higher dimensional analogues for Minkowskian simplices are studied in [237], [51], and [721]. There are also nice results on reduced polygons in normed planes (cf. [696] and [60]): For every side of a reduced polygon P the minimal width is attained in the direction perpendicular to it (i.e., orthogonal, in the Euclidean background metric, to the corresponding minimal-width strip). This property has no analogue for $n \ge 3$. If a convex polygon is reduced, then its central symmetral has to be a polygon circumscribed about a ball of the respective norm (i.e., each side of this polygon has to belong to some supporting line of that ball).

Geometric properties of general reduced bodies in normed planes were mainly obtained in [328] and [697]. These are general boundary properties, and also inequalities, approximation results and further natural generalizations of the analogous results for ${\mathbb {E}}^2$. Furthermore, in [971] Richter studied deeply the ratios of diameter and width of reduced and of complete convex bodies. Confirming a conjecture from [702], he showed that for $n > 2$ there exist reduced bodies of arbitrarily large ratio, whereas the ratio for complete bodies is bounded by $(n+1)/2$. As a consequence of these results, every normed space of dimension at least 2 contains reduced bodies that are not complete.

Staying with the idea of non-Euclidean geometries, we mention that research on spherical reduced bodies has also started. For example, in [698] the following results are derived: Every reduced spherical polygon is an odd-gon of thickness h at most $\frac{\pi }{2}$, and such a spherically convex odd-gon is reduced if and only if the projection of each of its vertices onto the great circle containing the opposite side belongs to the relative interior of that side, and the distance of this vertex from that side is h. An upper bound on the diameter of reduced spherical polygons is also given, with equality for regular spherical triangles. And in [699] it is shown that any smooth reduced spherical body is of constant spherical width.

Intersection Properties

An n-dimensional convex body K has the spherical intersection property (SIP) if it is the intersection of all balls whose center is from K and whose radius equals the diameter of K. In Euclidean space, the notions of (SIP) and constant width are equivalent; in Minkowski spaces this is no longer true, but (SIP) is still equivalent to completeness; see [316] and [238, p. 62]. If K is of constant width, let t(K) denote the smallest number of balls whose intersection is K. Soltan [1075] studied the number t(K), giving, e.g., necessary and sufficient criteria for $t(K) < \infty$ and constructing unit balls for even values in the plane. Groemer [457] used the (SIP) to study sets with unique completions in Minkowski spaces. And we repeat once more that Sallee [1001] extended Chakerian’s inradius–circumradius theorem from sets of constant width (cf. [231]) to sets with the (SIP). In [668], [789] and [774], relations between the notions of ball intersection (the intersection of all unit balls with centers from the given set), completion, (SIP) and constant width are investigated. (Note that a set has the (SIP) if it coincides with its ball intersection.) In the first paper the related weak spherical intersection property (WSIP) is introduced (for the Euclidean plane): a planar set S of diameter 1, say, has the (WSIP) if the intersection of all unit circles with center in S is of constant width. A finite example is the vertex set of a Reuleaux polygon. The least number of points (given in terms of the diameter graph) to be added to a finite planar set of diameter 1, say, such that the resulting set has the (WSIP), is determined. It would be very interesting to study (WSIP) sets also in higher dimensions and Minkowski spaces. In [789], the following further results for normed planes are shown: A convex body of diameter 1 satisfies the (SIP) iff it is of constant width; if a convex body has constant width, then it satisfies the related spherical hull property (defined via the intersection of all unit balls containing the given set). If a convex body and its spherical intersection are both of diameter 1 and it satisfies the spherical hull property, then this convex body has constant width. Further results from [789] refer to Reuleaux polygons and (WSIP) sets in normed planes. The paper [774] refers to applications of these notions in n-dimensional Minkowski spaces to Chebyshev sets and Chebyshev centers of bounded sets, thus giving also links to approximation theory and location science. In [788] new insights into representations of complete sets and (pairs of) sets of constant width are obtained, e.g., as vector sums of suitable ball intersections and ball hulls, a strongly related notion. Baronti and Papini [77] proved several properties combining completeness and the (SIP), also for infinite dimensions.

Curvature and Mixed Volumes

It turns out that classical results on principal radii of curvature (see [160, p. 128]), the surface area function and mixed volumes (cf. [1039, § 4.2 and § 5.1]) and, coming from this, a nice characterization of bodies of constant width via their surface area function (= first curvature measure of second kind) at opposite Borel sets on the unit sphere $S^{n-1}$ can be carried over to constant width sets in Minkowski spaces, using the tools of the so-called relative differential geometry discussed in [160, § 38] and [238, § 6], see also [522] and [523]. For a smooth convex body $\Phi \subset \mathbb {E}^n$ let $L({{\,\mathrm{\mathrm {bd}}\,}}\Phi , x)$ denote the canonical linear mapping of its tangent space ${{\,\mathrm{\mathrm {bd}}\,}}\Phi _x$ at $x\in {{\,\mathrm{\mathrm {bd}}\,}}\Phi$ into the tangent space of $\mathbb {S}^{n-1}$ at $u\in \mathbb {S}^{n-1}$, where x and u are connected by the usual Gauss map via parallel normals, with u as a unit outward normal of ${{\,\mathrm{\mathrm {bd}}\,}}\Phi$ at x. If for a smooth unit ball $\mathcal{G}$ in n-dimensional Minkowski space M, $\mathcal{G}_e$ denotes the tangent space of ${{\,\mathrm{\mathrm {bd}}\,}}\mathcal{G}$ at e having the same unit normal as ${{\,\mathrm{\mathrm {bd}}\,}}\Phi$ at x, then the linear map $J:{{\,\mathrm{\mathrm {bd}}\,}}\Phi _x \rightarrow \mathcal{G}_e$ can be defined by $J=L({{\,\mathrm{\mathrm {bd}}\,}}\Phi , x)^.L^{-1}({{\,\mathrm{\mathrm {bd}}\,}}\mathcal{G}, e)$ and is, as the canonical linear map of ${{\,\mathrm{\mathrm {bd}}\,}}\Phi$ into ${{\,\mathrm{\mathrm {bd}}\,}}\mathcal{G}$ via parallel normals, invertible with $J^{-1}=L({{\,\mathrm{\mathrm {bd}}\,}}\mathcal{G}, e)^.L^{-1}({{\,\mathrm{\mathrm {bd}}\,}}\Phi , x)$. Thus we may write $J=J(u)$, where u is the outward unit normal vector of ${{\,\mathrm{\mathrm {bd}}\,}}\Phi$ at x, and of ${{\,\mathrm{\mathrm {bd}}\,}}\mathcal{G}$ at e. The relative principal radii of curvature $\tilde{R}_1,\dots ,\tilde{R}_{n-1}$ of ${{\,\mathrm{\mathrm {bd}}\,}}\Psi$ at x are the reciprocals of the eigenvalues of J(u), and the corresponding relative principal directions are the eigenvectors of J(u). Following [160, p. 64], we write $\{\tilde{R}_1,\dots ,\tilde{R}_{\nu }\}$ for the $\nu$-th elementary symmetric function of the relative principal radii of curvature $\tilde{R}_1,\dots ,\tilde{R}_{n-1}$ and we let $F_\nu (\Phi , u)=\{\tilde{R}_1,\dots ,\tilde{R}_{\nu }\}$ at u be as above, see Section 11.2.1. With this notation, Chakerian [231] proved that if in addition $\Phi$ has constant width h, then $\tilde{R}_i(u)+\tilde{R}_{n-i}=h$, for every $u\in \mathbb {S}^{n-1}$, and in particular $F_1(\Phi , u)+F_1(K,-u)=(n-1)h$, $u\in \mathbb {S}^{n-1}$. Furthermore, we may define the relative mixed surface area as $S(\Phi , \dots ,\Phi ,\mathcal{G},\dots ,\mathcal{G}, E )$, where the convex body $\Phi$ appears r times, $\mathcal{G}$ appears $n-r-1$ times, and E is a Borel set in $\mathbb {S}^{n-1}$, see Section 12.2. In the same paper Chakerian proved that if $\Phi$ has constant $\mathcal{G}$-width then

$$\begin{aligned} S(\Phi ,\mathcal{G},\dots ,\mathcal{G}, E)+ S(-\Phi ,\mathcal{G},\dots ,\mathcal{G}, E)=2 S(\mathcal{G}, E) \end{aligned}$$

(10.2)

and, conversely, if the unit ball is smooth and (10.2) holds then $\Phi$ is of constant width. Hug [563] obtained further deep results in this direction, some of them also related to pairs of constant width in Minkowski spaces. In terms of difference bodies and for normed planes, the constancy of the sum of the radii of curvature of correspondingly opposite boundary points of a planar body of constant width was first observed by Vincensini [1152], and reproved by [1113, p. 31] and [929, Theorem (6.14)]. Also for normed planes, Chakerian [236] proved that an integral presented by the relative radius of curvature and the relative arc length element of a set K of constant width can be expressed by a double integral based on the number of diameters of K passing through the points of this set. For twice continuously differentiable convex curves in the plane, the four vertex theorem says that there are at least four vertices, i.e., points where the curvature has a stationary value. Heil [522] derived four vertex theorems for different definitions of curvature in normed planes, see also [523]. His results imply that in a normed plane any smooth curve of constant width has at least six vertices, see also [827]. Here we also mention the paper [66], in which all possible curvature types of smooth curves in normed planes are derived and classified. Based on this, several results (like Barbier’s theorem, or statements close to Theorem 11.3.1) are elegantly reproved, yielding also analogous statements for curves of constant anti-width (which refers to the width in the antinorm) if the anti-curvature radius (i.e., the inverse of the normal curvature) is used. A representation of k-th quermassintegrals of convex bodies of constant width in $\mathbb {E}^n$ in terms of mixed volumes due to Dinghas [296] was extended by Chakerian [235] to normed spaces. Finally we mention here that Guggenheimer (see [491, p. 327]) announced related results on sets of constant width, but more generally for gauges (where the unit ball is no longer centered at the origin).

Inequalities

The Blaschke–Lebesgue theorem for normed planes, saying that also there the Reuleaux triangle of width h has minimum area among all planar sets of constant width h, was proved by D. Ohmann and, independently, K. Günther (in both their dissertations, Marburg 1948); see also [898], [707], [929, pp. 15–16], [661], [230], and [410] for further approaches and modifications of this result. But in Minkowski planes Reuleaux triangles of given width h can have different areas. The particular type reaching the minimum (and obtained on independent, different ways in [898], [230], and [661]) can be constructed as follows: Let the Minkowskian unit vectors $r_1,r_2,r_3$ add up to o. Then one can form a triangle $a_1a_2a_3$ by suitable translates of the segments $or_i$ such that $a_1=o$ and $a_2, a_3$ are from the unit circle. Connecting $a_1$ and $a_2$, as well as $a_2$ and $a_3$, by corresponding boundary arcs of the unit ball, we get a figure of constant width having $a_1,a_2,a_3$ as boundary points and called by Ohmann general Reuleaux triangle. The announced analogue of the Blaschke–Lebesgue theorem says that for Minkowski planes among all figures of constant width $h > 0$ a general Reuleaux triangle has minimum area. Ghandehari [410] gave an optimal control formulation of this statement. Here we mention also the paper [1063], in which (similar to [661]) inequalities between several quantities are established. More precisely, given two quantities, the extremum values of a third one are established and the corresponding extremal bodies are determined. For example, Reuleaux polygons play a role there. And Sallee announced in [999] that the Minkowskian analogue of the Firey–Sallee theorem, saying that among all Euclidean Reuleaux polygons of width $h > 0$ and having $m \ge 3$ vertices, the regular one has maximum area, can be proved on the same lines as the Euclidean subcase. In [1197] and [785] it is shown that the ratio of the area of the unit ball to that of a Reuleaux triangle of width 1 lies between 4 and 6, and in [785] several further inequalities on areas of Reuleaux triangles and a new proof of Barbier’s theorem are given. Also the paper [965] refers to Reuleaux triangles as extremal figures. Castro Feitosa [336] used extremal properties of figures of constant width in normed planes to extend results of Scott [1052] from the Euclidean case to Minkowski planes. More precisely, he used that among all convex figures in normed planes with given diameter and circumradius the sets with largest thickness, perimeter, inradius, and area are of constant width. More inequalities between inradius, circumradius, diameter, and related quantities of convex bodies in Minkowski spaces yielding in some cases sets of constant width as extremal cases are discussed in [313], [1063], [144], [1082], [709], and [710]. The Rosenthal–Szasz inequality between perimeter and diameter of a planar convex figure was extended in [65] to Radon planes, and an analogue (considering the diameter in the antinorm) was also proved there. Here we mention also the papers [53], [718], [56], [222], and [626]. For example, in the latter one the authors give an upper bound on the area of a constant width figure in terms of the Minkowskian arc length of its pedal curve and further quantities. An analogous result in terms of its width, the minimal radius and the arc length of its pedal curve was derived in [224]. This bound is reached iff the constant width figure is homothetic to the pedal curve of the isoperimetrix of the normed plane under consideration. Ghandehari proved in [411] that in n-dimensional Minkowski space the volume of the polar reciprocal of a given convex body of constant width 2 is bounded from below by the volume of the dual of the unit ball; equality holds iff this constant width body is the unit ball. Another type of result is derived in [754]: For a finite point set X with n-dimensional convex hull P, the points $x_i, x_j \in X$ are called antipodal if there are different parallel supporting hyperplanes $H', H''$ of P with $x_i \in H'$, $x_j \in H''$. If the considered space is Minkowskian, one might ask for the number of pairs in X whose Minkowski distance is maximal. This number is not larger than the number a(X) of antipodal pairs in X, and if the polytope P is of constant Minkowski width, then equality holds. In [754] several upper bounds on a(X) are derived.

Projections

We come now to Minkowskian analogues of results discussed in the notes to Chapter 13. In Minkowski spaces we say that a convex body K is of constant brightness if the brightness of K (measured as in the Euclidean case) is proportional to that of the unit ball. Chakerian [231] showed that if a 3-dimensional convex body K is of constant brightness and its unit ball have $C^2$ boundary with everywhere positive curvature, then K is a ball. More generally, Petty (see [930] and [931]) defined the brightness of K at $u \in S^{n-1}$ in an n-dimensional Minkowski space as the minimal Minkowski cross-section area of the cylinder $K+l$, where l is a 1-subspace of direction u. In [930] and [931] he derived results on bodies of constant brightness and constant width, e.g., as extremal cases of certain inequalities. Chakerian [231] studied also sets of constant k-girth in Minkowski spaces, and Petty [931] studied bodies of constant Minkowskian curvature.

Discrete Geometry

Following Loomis [730], a set K three-covers the set L if $L \subseteq \{a_1,a_2,a_3\} + K$ for some three points $a_1,a_2,a_3$. Having this notion, he showed that a Reuleaux triangle of width h three-covers any figure of constant width h, and that if the unit ball is a centrally symmetric octagon, then every figure of constant width three-covers every other set of the same constant width. Further on, the dissertation [730] contains also Helly-type theorems for sets of constant width in normed planes. Motivated by a question of Hammer, Sallee [997] investigated bodies of constant width in association with lattices. Defining Reuleaux polygons of width 1 as finite intersections of (properly chosen) translates of the unit ball and saying that a set S avoids another set X if the interior of S does not meet X, he proved the following statements for strictly convex Minkowski planes: Every set of maximal constant width avoiding a square unit lattice L is a Minkowskian Reuleaux triangle P where each of the three open “edges” of P contains at least one point from L. And if the lattice L is replaced by a locally finite family X of convex sets in an arbitrary normed plane, then the correspondingly maximal sets are Reuleaux polygons all whose open “edges” contain points from X. Our next theme is the famous partition problem of Borsuk. Surveys on this problem, also referring to Minkowski spaces, are [488], [151, Chapter V], and [960], see also [272, D 14], [150], [154]. This problem is closely related to bodies of constant width and completeness (see, e.g., [151, Chapters V and VIII]). A first investigation for $n = 2$ was done by Grünbaum [485]; he showed that if the unit ball is not a parallelogram, then any set of diameter 1 can be covered by three balls of smaller diameters. The so-called Borsuk number can be defined as follows. Let F be a bounded set in a Minkowski space having diameter h. What is the smallest integer k such that F is the union of k sets each having diameter $< h$? Denoting this smallest number by $a_B(F)$, Boltyanski and Soltan [155] proved for $n = 2$ that $a_B(F) \in \{2,3,4\}$, where $a_B(F) = 4$ iff the unit ball B is a parallelogram and the convex hull of F is homothetic to B. And $a_B(F) > 2$ holds iff one of the following two conditions is satisfied: (A) There is a unique completion of F to a figure of constant width h. (B) For any two parallel supporting hyperplanes of this constant width completion at least one has nonempty intersection with F. In [575] some conditions are derived under which an n-dimensional body of constant width in the sense of the norm can be presented as union of $n + 1$ subsets of smaller diameter than K has. Also related to (unique) completions of compact sets, in [788] some new results on Borsuk numbers of sets of constant width in normed spaces are derived. We finish this subsection by some results on in- and circumscribing figures. It is well known that a hexagon (regular in the considered planar norm) can be inscribed in any Minkowski circle. Thompson [1124, Chapter 4] gives a nice discussion of related results, including a way how this theorem can be applied to construct curves of constant width, in particular also Reuleaux triangles. As a special case of a result of Doliwka [297] (conjectured by Lassak [691]), the following result on normed planes holds: any planar figure of constant width 1, say, has an inscribed pentagon whose vertices are in at least unit distance to each other. Related results can be found in [683] and [694]. Spirova [1086] extended a result of Chakerian to normed planes: if a planar convex body can be covered by a translate of a Reuleaux triangle, then it can be covered by a translate of any convex body of the same constant width. In [790] universal covers in Minkowski planes are investigated; it turns out that a related covering property of constant width sets is needed in the proofs.

Related Concepts

As we discussed it already in Section 7.3, Maehara [747] defined two convex bodies $K_1, K_2$ in $\mathbb {E}^n$ to be a pair of constant width if $K_1 + (-K_2)$ is a ball. Analogously, Sallee [1002] defined $\{K_1, K_2\}$ to be a pair of constant width in Minkowski spaces if the sum of their support functions in the form $h(K_1,u) + h(K_2,-u)$ is itself the support function of a Minkowskian ball. He proved that $K_1$ is a summand of the unit ball iff there is a $K_2$ such that $\{K_1,K_2\}$ is a pair of constant width in Minkowskian sense. Also his generalization of Chakerian’s theorem on in- and circumradius of constant width sets, given in [1001], can be formulated in terms of pairs of constant width. The authors of [253] characterized pairs of planar sets of constant width in terms of the Fourier coefficients of their radii of curvature. Weakly related is also the paper [33]. Rodriguez Palacios [979] pointed out that results on summands of Banach spaces may be interpreted in terms of sets of constant width, see also [922]. In view of multiples with scalars, Minkowski sums, and suitable combinations thereof, the family of all convex bodies in $\mathbb {E}^n$ forms an abelian semigroup with scalar operators. Having such an algebraic structure and the Hausdorff metric in mind, Ewald and Shephard [326] introduced an equivalence class structure of the subclass of bodies of constant width, yielding an incomplete normed linear space, and they mentioned that this idea can (due to a suggestion of Grünbaum) easily be extended to bodies of constant width in Minkowski spaces. Sorokin [1083] gave such extensions, even for nonsymmetric unit balls (which, on the other hand, have to be smooth and strictly convex). Taking the minimum width of certain representatives as a metric (in the abovementioned space), Lewis [723] showed that then a conjugate Banach spaces with complete norm is obtained. Related is also the paper [324]. Finally in this subsection, we mention that relations between the concept of bodies of constant affine width (see Section 11.6) and Minkowski geometry were studied in [542] and [99]; the background is given by typical methods from relative differential geometry in the sense of [160] (see Section 38 there).

Further papers of Hirakawa related to affine relative differential geometry and considering also curves of constant width within this framework are [543] and [544]. We continue with mentioning results about rotors in normed planes. Ghandehari and O’Neill [414] derived inequalities for the self-circumference of rotors in equilateral triangles and figures of constant width. Hereby these self-circumferences are measured by taking these rotors or figures of constant width themselves as gauge figures or unit circles of Minkowski planes. The inequalities compare their areas and mixed areas (using also the polar reciprocal) with the self-circumference. Also higher dimensional results are given there. Sorokin [1083] investigated classes of convex bodies which can roll in Minkowskian spheres. We finish with a link to Zindler curves in normed planes. Our subsection 5.4.1 clearly shows that this type of curves is closely related to constant width curves in the Euclidean plane, and their analogues also in higher dimensions. Since this notion can also be carried over to normed planes (see [798] and [799]), it certainly yields an interesting field of further research.

Constant Width in Other Non-Euclidean Geometries

The concept of constant width was also extended to non-Euclidean geometries different to Banach space geometry. There exist results mainly for hyperbolic and spherical spaces, and also for the complete concept of spaces of constant curvature. In addition, also results on constant width sets in Riemannian manifolds and in further frameworks exist; we try to give an overview to results that are known. Again (as in Minkowski spaces) one can observe that already the definitions of notions give an interesting variety of possibilities.

Hyperbolic Geometry

Santaló [1015] defined geodesic convex curves of constant width in the hyperbolic plane by means of supporting and transversal geodesics, and based on that he derived already extremal properties of analogues of Reuleaux polygons and an analogue of Barbier’s theorem, see also [1020]. Continuing this, Fillmore [350] defined curves of constant width as horocycle convex curves having constant distance between every two supporting horocycles with the same center at infinity (where the distance is measured along certain geodesics). Fillmore showed that not all curves of the same constant width have the same length (as it is the case with the concept developed in [1015]); it is also shown how the Euclidean borderline case naturally leads to Barbier’s theorem. Using different methods, Leichtweiss [715] defined a closed convex curve to be of constant width h if its supporting strip region has the same width h for any direction (here a strip region is defined as a region, in the hyperbolic plane as well as on the sphere, bounded by two curves which have constant distance from a given geodesic line). Continuing [1015] and [350], Leichtweiss also derived a version of Barbier’s theorem, and based on approximation arguments for Reuleaux polygons and a careful analysis of their area he succeeded with a hyperbolic version of the Blaschke–Lebesgue theorem yielding analogues of Reuleaux triangles as extremal figures. Many of his results refer also to spherical space. Another way to extend Barbier’s theorem to complete, simply connected surfaces of constant curvature k is given in [34]. The author underlines that the case $k > 0$ is due to Blaschke [130] and $k < 0$ to Santaló [1015], but that his approach (using tools from differential geometry) is new and unifies these results. Coming from an isoperimetric inequality referring to horocycle convex curves of the hyperbolic plane (see also above), Pinkall [937] proved an inequality for curves of constant width, elegantly using an ingenious map of the horocycle convex curves on the centrally symmetric curves of the affine plane. He also reproved results from [1015] and [1020], such as Barbier’s theorem. Also [35] refers to such problems in the hyperbolic plane. A simple closed curve has constant width h if for each point from it the maximum hyperbolic distance to other curve points is h. Using a definition of Reuleaux triangle exactly as in the Euclidean case, the author of [35] confirmed that among all piecewise regular curves of the same constant width the Reuleaux triangle encloses the smallest area, and also this is obtained via new differential-geometric methods, giving an alternative proof without invoking the isoperimetric theorem that the circle encloses the largest area among all curves of the same hyperbolic constant width. This author continued in [36] with presenting a parametrization of curves of constant width in the hyperbolic plane, giving also examples for their usage. As a central result it is shown that each curve parametrized this way can be uniformly approximated by analytical curves of constant width, furthermore used for extending known results. In [582], Jerónimo-Castro and Jimenez-Lopez characterize the hyperbolic disk among constant width sets.

It is well known that a planar convex body in $\mathbb {E}^2$ with the property that from every point outside of it the two respective tangent segments have equal length, has to be a Euclidean ball. The authors of [585] investigate various related problems and obtain a nice characterization of constant width sets in the hyperbolic plane working explicitly with the metric of the upper half-plane model. In [387] the relation between asymptotic values of the ratios area/length and diameter/length of a sequence of convex sets expanding over the whole hyperbolic plane is investigated. It is shown that diameter/length has its limit value between 0 and $\frac{1}{2}$, in strong contrast to the Euclidean situation where the lower bound is $\frac{1}{\pi }$, holding iff the convex sets have constant width. In [388], convex bodies in spaces of constant curvature are investigated, and several interesting and useful expressions for the measures of lines and planes intersecting such bodies are obtained. Since also here a natural definition of width comes into the game, bodies of constant width become interesting in this framework. Via a characterization of the ball by some inequality in terms of body volume, ball volume and ball area, finally a complete system of related inequalities for constant width bodies is obtained.

Glasauer [420] derived a local Steiner formula and showed that this formula can be applied to bodies of constant width in Euclidean, spherical, and hyperbolic space. Dekster [289] introduced the concept of completeness for compact sets in hyperbolic, Euclidean, and spherical space, showing that in the first two cases the notions of completeness and constant width coincide. For getting this coincidence in the spherical case, convexity has to be assumed. Moreover, the author also constructed nice examples showing how strong complete sets of diameter larger than $\frac{\pi }{2}$ can differ from bodies of the same constant width. Studying isoperimetric problems in hyperbolic, Euclidean, and spherical surfaces referring to regions between constant-curvature curves, Simonson [1066] showed that certain minimizing sets cannot occur in regions of constant width.

As an extension from a ball-shaped basic region to arbitrary domains, the Apollonian metric is a generalization of the hyperbolic metric, and relations between these concepts were investigated in [567]. It turns out that if these domains are complements of sets of constant width, then the metric has special interesting properties, see [568], [569], and [165].

Spherical Space

Before we discuss references referring only to spherical space, we underline that the above-cited papers [130], [34], [289], [420], [715], and [388] cover (besides results holding in hyperbolic geometry) also analogous results for the spherical case.

Two early contributions are [130] and [1012]. Properties of constant width sets on the unit sphere are discussed, and an extremal property of the spherical Reuleaux triangle is proved. The paper [1014] continues Blaschke’s paper [130], characterizing spherical curves of constant width in two inequalities in terms of minimal width and diameter. The paper [291] combines transnormality and constant width sets in spherical n-space. Considering their inner and outer parallel sets (which are 2-transnormal topological ($n-1$)-spheres), the authors prove that if C is a spherical body of constant width $(\frac{\pi }{2}) + a, 0< a < \frac{\pi }{2},$ then C is smooth and contains a body $C'$ of constant width $(\frac{\pi }{2}) - a$ such that C is the union of the spherical caps of radius a whose centers are from $C'$. In analogy to quermassvectors of convex bodies in Euclidean space (see [1035]), Arnold [38] introduced quermassvectors for strictly convex subsets of the 2- and 3-dimensional spherical space. These spherical quermassvectors can, after renormalization, be interpreted as curvature centroids, and some properties of them are proved. In particular, vector-valued Steiner formulas for the quermassvectors of outer parallel bodies are derived. Also Lassak [699] studied n-dimensional spherical sets of diameter at most $\frac{\pi }{2}$ and defined in a natural way spherical bodies of constant width and, in addition, also reduced sets. He showed that then all bodies of constant width are, like also any such spherical odd-gon for $n = 2$, reduced sets, and that any reduced and smooth spherical body is of constant width. Further results on spherical reduced sets are presented in [879]. Recently, Lassak and Musielak (see [698], [703], and [704]) obtained new results in reduced bodies in spherical geometry, see also [879]. Spherical versions of Reuleaux polygons of diameter $< \frac{\pi }{2}$, also extendable to the hyperbolic plane, are given in [673]. The paper [703] contains results of the following type, again referring to the n-dimensional spherical space: For a certain definition of width, the authors introduce bodies of constant width h, proving that their diameter is h and that for $h < \frac{\pi }{2}$ these sets are strictly convex. The concepts of “constant width” and “constant diameter” in this framework are compared, leaving open the question whether any spherical body of constant diameter $h < \frac{\pi }{2}$ is a body of constant width h. The concept of completeness in spherical spaces is investigated in [318], [289], and [1180]. Among other results, one can find there the theorem that for sets of diameter smaller than $\frac{\pi }{2}$ the notions of completeness and constant width coincide. In [34] the following nice characterization of the circle within the family of constant width curves (for the Euclidean case due to Hammer and Smith [510]) is extended to the 2-dimensional spherical space: if each two points on a curve of constant width h, having spherical distance h from each other, divide the curve into two arcs of equal length, then the curve is a circle.

Pottmann ([951] and [954]) showed that spherical curves of constant width $\frac{\pi }{2}$ are interesting in view of spherical motions. Related to the problem of finding the maximal length of steepest descent spherical curves for quasi-convex functions satisfying suitable constraints, the spherical Reuleaux triangle occurs in [729] as extremal figure. The following result from [114] holds on the 3-sphere: the volume of any spherical convex body of constant width h is minimal among all bodies of constant width h iff the polar body of constant width $\pi - h$ has minimal volume among all bodies of constant width $\pi - h$. Representing results on systems of inequalities due to Minkowski one can use spherical geometry, and a related characterization of spherical sets of constant width $\frac{\pi }{2}$ is given in [140]. Further on we refer in this subsection to [963].

On Manifolds and Further Non-Euclidean Concepts

In a series of papers, Dekster worked successfully on the extension of the concept of constant width to manifolds, and one should also mention the earlier related papers [894] and [895]. In [285], Dekster investigated compact, geodesically convex subsets of an n-dimensional Riemannian manifold such that a geodesic segment from them never contains a pair of conjugate points. Such a set is called a body of constant width h if for each of its boundary points any normal geodesic chord emanating from it has length h and is a maximal geodesic chord of the set with this endpoint. The author confirmed that various standard properties of Euclidean sets of constant width also hold for their analogues in Riemannian manifolds, and others do not. In [284] he continued by showing that each incenter of such a constant width body is a circumcenter and, conversely, that there is a unique point equal to both, under certain curvature conditions. Results on in- and circumradii are given, too. In [286] the concept of width of a convex body in a Riemannian manifold is newly introduced (in other papers the author is not using the notion of “width” explicitly, but prefers some analogue of “constant diameter”), and it is confirmed that such a convex body is of constant width iff also this type of width is constant. Two extensions of the characterization via double normals are given in [288], and further possible characteristic properties of constant width sets in manifolds are discussed, see also [287].

In this last paragraph we discuss (somehow widespread) results on the concept of constant width in further non-Euclidean geometries. The paper [1] contains a deep study of many basic notions from convexity in complex vector spaces-among them also sets of constant width. We continue with a special Cayley–Klein geometry. Namely, Strubecker [1102] studied Zindler curves and constant width curves in isotropic geometry, transforming the framework suitably into isotropic space (thus using then a spatial approach to planar results). The papers [644] and [613] refer to spacelike curves of constant width in “the other Minkowski geometry”, i.e., in Minkowskian spacetime 4-space, whereas [900], [646], [1207], and [643] refer to the 3-dimensional analogue. More precisely, in [900] and [644] differential equations characterizing such curves are obtained, and in [613] Frenet-like formulae are developed. In [646] special types of such curves are investigated, and it is shown that the total torsion of a closed spacelike curve of constant width is zero. Properties of double normals of these curves are obtained in [644], in [643] these curves are investigated according to the Bishop frame, and [1207] contains a study of the correspondingly dual curves, sitting in the dual Lorentzian 3-space. In these publications also more related references can be found. We should also mention the paper [682] in which some concept of constant width in Möbius geometry is developed. In an abstract sense, tolerance classes can be introduced in a natural way in any metric space. In [658] relationships between two such classes and bodies of constant diameter in this model are established. Our last reference cited here refers to relations between lattice theory and metric spaces. Namely, in [615] lattices are “normed” such that they yield metric spaces with a well-defined distance function. The author establishes necessary and sufficient conditions for a general metric space to be a space associated with a normed lattice in this way, and it turns out, that sets of constant width (defined with the help of that distance function) are useful for such investigations.

Exercises

• 10.1. Let $H_1$ and $H_2$ be parallel support hyperplanes of the ball ${\mathcal {G}}(x, r)$, and let $x_1 \in H_1$ and $x_2 \in H_2$. Prove that $x_1x_2$ is a bridge of the strip ${{\,\mathrm{\mathrm {cc}}\,}}(H_1\cup H_2)$ if and only if there are points $y_1 \in H_1\cap {\mathcal {G}}(x, r)$ and $y_2 \in H_2\cap {\mathcal {G}}(x, r)$ such that $x_1x_2$ is parallel to $y_1y_2$.

• 10.2. Prove that a line L and a hyperplane H in the Minkowski space M are ${\mathcal {G}}$-perpendicular if the line $L_0$ through the origin parallel to L has the property that the unit ball ${\mathcal {G}}$ has the support hyperplanes through $L_0\cap {{\,\mathrm{\mathrm {bd}}\,}}{\mathcal {G}}$ parallel to H.

• 10.3. Suppose the unit ball ${\mathcal {G}}$ of the Minkowski space M is strictly convex. Prove that if two lines are ${\mathcal {G}}$-perpendicular to a hyperplane, then they are parallel.

• 10.4. Suppose the unit ball ${\mathcal {G}}$ of the Minkowski space M is smooth. Prove that if two hyperplanes are ${\mathcal {G}}$-perpendicular to a line, then they are parallel.

• 10.5. Suppose the unit ball ${\mathcal {G}}$ of the Minkowski space M is strictly convex, and let $\Phi \subset M$ be a body of constant ${\mathcal {G}}$-width. Prove that $\Phi$ is strictly convex.

• 10.6. Prove Theorem 10.2.4 assuming that $\phi _1$ and $\phi _2$ are smooth convex bodies.

• 10.7. Prove that every ${\mathcal {G}}$-diameter is a ${\mathcal {G}}$-binormal.

• 10.8. Prove that any two parallel ${\mathcal {G}}$-binormals have the same ${\mathcal {G}}$-length.

• 10.9. Suppose that $\Phi$ has a ${\mathcal {G}}$-binormal of ${\mathcal {G}}$-length h in every direction. Prove that $\Phi$ has constant ${\mathcal {G}}$-width h.

• 10.10**. Construct a 3-dimensional ${\mathcal {G}}$-complete body which is not a body of constant ${\mathcal {G}}$-width.

• 10.11. Let $\phi$ be a centrally symmetric convex set in the Minkowski space M with unit ball ${\mathcal {G}}$. Suppose the origin is the center of $\phi$ and the chord rs is a ${\mathcal {G}}$-diameter of $\phi$. Prove the following two statements.

  • (a) The chord $r(-r)$ is a ${\mathcal {G}}$-diameter of $\phi$.

  • (b) The chord pq through the center of $\phi$ parallel to rs is a ${\mathcal {G}}$-diameter.

• 10.12. For a fixed point p on the unit circle ${\mathcal {G}}$ of a Minkowski plane M and a variable point x, which is moving on ${{\,\mathrm{\mathrm {bd}}\,}}{\mathcal {G}}$ from p to $-p$, prove that the ${\mathcal {G}}$-length of the chord px is not decreasing.

• 10.13. Prove that if $\Phi \subset M$ has constant ${\mathcal {G}}$-width in the Minkowski plane M with unit disk ${\mathcal {G}}$, then every chord pq of $\Phi$ splits the body $\Phi$ into two compact sets such that one of them has ${\mathcal {G}}$-diameter equal to the ${\mathcal {G}}$-length of pq.

• 10.14. A strip is called ${\mathcal {G}}$-regular if the corresponding parallel support ${\mathcal {G}}$-strip is regular. Prove that the ${\mathcal {G}}$-diameter of $\Phi$ is the maximum of the ${\mathcal {G}}$-widths of all ${\mathcal {G}}$-regular support $\Phi$-strips.

• 10.15. Let $H_1$ and $H_2$ be parallel hyperplanes of an n-dimensional Minkowski plane M with unit ball ${\mathcal {G}}$ and suppose $d_M(H_1,H_2)=h$. Let $x_i\in H_i$, $i=1,2$. Prove that there exist parallel $(n-2)$-dimensional affine planes $L_1, L_2$ such that $x_i\in L_1\subset H_i$, $i=1,2$, and with the property that it is possible to parallel-rotate $H_1,H_2$ in such a way that the ${\mathcal {G}}$-width of the resulting strips is always smaller than or equal to h.

• 10.16. Let ${\mathcal {G}}$ be a centrally symmetric convex body in Euclidean n-space $\mathbb {E}^n$ and let p and q be two points with the property that they both can be covered with a translated copy of ${\mathcal {G}}$. The ${\mathcal {G}}$-interval determined by this pair of points is the intersection of all translated copies of ${\mathcal {G}}$ that contain p and q. A closed set $\phi$ is called spindle ${\mathcal {G}}$-convex if it has the property that, given a pair of points p and q in $\phi$, the ${\mathcal {G}}$-interval that they determine is also in $\phi$.

  • (a) Let $\phi$ be a strongly ${\mathcal {G}}$-convex set. Prove that $\phi$ is spindle ${\mathcal {G}}$-convex,

  • (b) Is every spindle ${\mathcal {G}}$-convex a strongly ${\mathcal {G}}$-convex set? Under what conditions on ${\mathcal {G}}$ is this true?

• 10.17*. Prove that balls and simplices in every dimension are generating sets. Prove that all 2-dimensional convex sets are generating sets.

• 10.18. Prove that the intersection of all spindle $\phi$-convex sets containing the compact set V is $\phi \sim (\phi \sim V)$.

• 10.19. For every direction u, let $\Gamma (\phi , u)$ be the maximum length of a chord of the convex body $\phi$ in the direction u. Prove that $\Delta (\phi )= {{{\,\mathrm{\mathbf {min}}\,}}}_{u} \Gamma (\phi , u).$

• 10.20. Give an example of a convex body which has constant ${\mathcal {G}}$-minimum width but is not a body of constant ${\mathcal {G}}$-width.

• 10.21. Let F be a parallelogram. Prove that $\beta _F(F)=4$.

• 10.22. Let ${\mathcal {G}}$ be a square, and F be a circle. Prove that $\beta _{\mathcal {G}}(F)=2$.