**Completeness and Completions**

A bounded subset of \(\mathbb {E}^n\) is called *complete* (or *diametrically complete*) if it cannot be enlarged without increasing its diameter. One can define this type of sets with the same formulation within the class of convex bodies and will get the same family of sets. It turns out that in Euclidean space the classes of complete sets and of bodies of constant width coincide (this is the so-called Meissner Theorem), see Theorem 7.2.1. For a convex body in \(\mathbb {E}^n\), any complete set containing it and having the same diameter is called a *completion* of that body. Furthermore, each convex body has at least one completion, which is the Theorem 7.2.2 of Pál. For the history of and early contributions to these notions we refer to [160, Section 64]. Meissner’s Theorem was derived in [817] for \(n = 2\) and \(n = 3\), and for arbitrary dimension in [586]. It can be used to show that a convex body is of constant diameter if it is of constant width, see [964]. And Pál’s Theorem was derived in [909] for \(n = 2\) (see also [966]), and in general by Lebesgue [708]. Based on ideas from [966], also Bückner [192] proved Pál’s Theorem (cf. also [193] and [99]), and Sallee [998] gave another proof in view of the spherical intersection property. Namely, he showed that the completion of the convex body *K* is an intersection of balls of the same diameter whose centers are taken from *K* and some countable, everywhere dense set in \(\mathbb {E}^n\). See the proofs of Theorems 7.2.4 and 10.3.2. This construction he used already in [997] to obtain new planar sets of constant width from old ones, replacing old boundary parts by certain circular arcs. Closely related to the concept of completeness in the plane, Rademacher and Toeplitz [957, § 20b], Mayer [806], and Bückner [192] found exact criteria for planar arcs to be boundary parts of 2-dimensional sets of constant width, and in [192] also 3-dimensional analogues have been verified. Related are also the papers [1048] and [389], the latter again only discussing the planar case. For Minkowski spaces, such investigations were done by Sallee in [1003], see Section 10.3 and the Notes of Chapter 10 below. Constructing a completion of *K*, say, by an increasing (regarding inclusion) sequence of sets, Scott [1052] showed that all of these sets must lie inside the circumsphere of *K*, yielding the result that any convex body *K* has a completion with the same circumradius as *K*, see Section 14.4. This containment result was also obtained by Vrećica and even extended to normed spaces, see [1159]. Considering not only the diameter of *K* in this framework, but also other quantities (such as minimum width, in- and circumradius, etc.), Groemer developed in [456] a general concept to look for related extremal bodies, regarding minima and maxima. He derived a general existence theorem for corresponding maximal and minimal sets and applied this also to extremal sets with respect to packing and covering densities. Regarding *minimal sets*, we have to mention that this concept inspired Heil to define the notion of reducedness (cf. [524]), motivated also by the isodiametric problem (asking for volume-maximizers for given diameter). Namely, he asked for the “dual” of this problem, i.e., for the convex bodies which are volume-minimizers for fixed given minimum width. Such sets have to be reduced, see our representation in this chapter and also Heil’s problems posed in the problem collection [481], as well as [272, A 18]. In [457] Groemer studied volume-maximizers in the family of all completions of a convex body, and he investigated also completions in Minkowski spaces. Replacing volume by mean projection measures, he further proved the existence and uniqueness of maximal completions having prescribed symmetry properties, thus extending related results of Schulte [1048] and Schulte and Vrećica [1049] (see also [1050]) to normed spaces. Another contribution to the preservation of symmetry properties when completing a convex body is [982]. It is shown there that any bounded *n*-dimensional set having a symmetry group *G* and diameter 1, say, is contained in a body of constant width 1 having the same symmetry group *G*, see also again [1048] and Section 7.3. In [747], Maehara introduced a completion operation for convex bodies in \(\mathbb {E}^n\) (see Section 7.5 and the Notes of Chapter 10), and in [748] he showed that by this procedure several known results on convex bodies can be reobtained, but in a way that their derivation does not depend on Blaschke’s Selection Theorem, see also the part “Pairs of constant width” in these notes.

By Falconer [331] a point *p* of a set of diameter 1 is said to be *singular* if there exist two distinct points of this set having distance 1 to *p*. Falconer proved that every compact subset *S* of \(\mathbb {E}^n\) of diameter at most 1 is contained in a set *X* of constant width 1 such that *S* and *X* have the same set of singularities. This yields a related approximation procedure (taking into consideration the structure of these singularities) within the family of sets of constant width; similar results were derived by Schulte in [1048]. A nice characterization of plane sets with *unique completion* was found by Boltyanski [146]: This property holds if the respective sets cannot be covered by two sets of smaller diameter each, thus giving a basic result on Borsuk partitions in two dimensions. An extensive discussion of the role that sets with unique completion and constant width sets play for results around Borsuk’s partition problem is given in Chapter 5 of [151], see also the related research problems in Chapter 8 there. Parts of the representation of Borsuk’s problem in normed spaces given in [151] are strongly based on the papers [485], [316], and [155], see also [488], [272, D 14], and [154]. Although mainly referring to normed spaces, the paper [890] contains also results on the Euclidean situation, e.g., about sets with unique extension to bodies of constant width, see the Notes of Chapter 10. Further results in this direction are given in [651] and [652]. For example, in the second paper it is shown that a planar convex body has unique completion if for any non-diametral chord of it there exists a diametral chord not meeting the interior of this chord; further results refer to Borsuk numbers (see Section 15.3) of convex bodies all whose diametral chords are in special positions. Still regarding Borsuk’s problem we refer also to A 27 in [272], where the following problem is posed: What is the largest possible diameter of the set *M* of midpoints of all diametral chords of a set of unit constant width in \(\mathbb {E}^n\), taken over all such sets? In particular, is there a simplex containing *M* and contained in *K*? A positive answer would imply an affirmative answer of Borsuk’s problem. In view of approximation procedures, Stefani (see [1097], [1091], and [1093]) studied measure-theoretic questions related to complete sets. And in [592] (continuing investigations from [591]) completions of regular simplices in \(\mathbb {E}^n\) are characterized via Minkowskian measures of asymmetry.

Following Eggleston [310], we say that a convex body *K* in \({\mathbb E}^n\) of diameter *h* has the *spherical intersection property* if *K* is the intersection of all balls *B*(*x*, *h*) of radius *h* with center \(x \in K\). It is easy to see that any body of constant width *h* has this property, and via completeness (see Chapter 7) also the converse is trivially shown. Hence a convex body *K* has the spherical intersection property if it is of constant width (we note that this equivalence no longer holds in Minkowski spaces, see Chapter 10). In papers like [806], [194], [127], [81], and [671] the relationships between constant width sets and sets which are convex in the sense of certain generalized convexity notions (like over-, super-, or hyperconvexity), the so-called adjoint transform, and the concept of ball polytopes is investigated. Going further to notions like ball hulls and ball intersections, one sees that these concepts are especially interesting in Minkowski geometry, see Sections 7.3 and 10.4. The same holds true for the concept of pairs of constant width, see [1001] and [747]. Here we have to refer also to the Notes of Chapter 4.

**Pairs of Constant Width**

Maehara [747] defined *pairs of constant width* as pairings (*X*, *Y*) of compact, convex sets whose sum \(h(X, u)+h(Y,-u)\) of support functions for all directions *u* equals a constant \(r > 0\), thus yielding that \(X + (-Y)\) is a ball. He proved that (*X*, *Y*) is such a pair if the intersection of all balls of radius *r* with center in *X* equals *Y* and, vice versa, *X* is obtained analogously from *Y* (examples of such pairs are constructed, too). In addition, Maehara [747] showed that the insphere of *X* and the circumsphere of *Y* are concentric (as they are in the usual constant width situation \(X = Y\)). We also refer to Section 7.3.2 and also to the part “completeness” of Section 10.7.

Earlier investigations in this direction are due to Valette [1141]. He also obtained (as a corollary) that for any compact, convex set *X* with twice continuously differentiable support function one can find a convex body *Y* of constant width such that \(X+Y\) is centrally symmetric. Sallee [1002] extended the notion of pairs of constant width to that of *S*-pairs (useful also for Minkowski spaces), meaning that \(h(X, u)+h(Y,-u)\) yields the support function of a prescribed centrally symmetric, convex body *S* (instead of a usual ball). Among other results he showed that *X* is a Minkowski summand of *S* if there is a set *Y* such that (*X*, *Y*) is an *S*-pair. The same author continued these considerations in [1001], involving also circumradii, inradii, and the spherical intersection property. The results of [1002] were extended in [167] replacing *S* by a set which, in general, is nonsymmetric (thus referring to constant width sets for gauges, which are more general than norms). Let *S* be a nonempty compact, convex set in the plane (not a singleton), and for each integer *k* let \(F_n\) be the set of points each having at least *k* farthest points in *S*. Among other things, the authors of [292] proved that \(T_3\) is countable and \(T_2\) is contractible to the circumcenter of *S*. For \(r>0\) let \(C_r\) be the set of points whose distance to a farthest point in *S* is *r*; \(C_r\) is a strictly convex curve in case *r* is larger than the circumradius of *S*. The authors of [292] were unaware of the fact that the boundary of *S* and the curve \(C_r\) form the boundaries of a pair of constant width as defined in [747]. Also unaware of [747], the authors of [253] characterized pairs of plane constant width sets in terms of the Fourier coefficients of their radii of curvature. Various results for such pairs are proved, and also a generalization of Barbier’s Theorem due to Maehara [747] is reproved in [253]. Aitchison [6] calls two convex bodies equivalent if the ratio of their widths for equal directions is constant over all directions. He applies this notion to suitable parallel pairs of sections of strictly convex bodies in 3-space, concluding that the boundary of these bodies has to consist of finitely many boundary pieces of ellipsoids. (Also a characterization of ellipsoids due to Blaschke is generalized in this paper, but this is more related to our Section 2.12.)

**Reduced Bodies**

The definition of reduced bodies, going back to Heil [524], can somehow be considered as a dualization of that of complete sets. Recall that the latter notion can be defined as follows: A convex body in \({\mathbb E}^n\) is said to be *complete* (i.e., in \({\mathbb E}^n\) of constant width) if any proper superset of it has larger diameter. Replacing superset by subset and diameter (= maximal width) by thickness (=minimal width), we have on the other hand: a convex body in \({\mathbb E}^n\) is called *reduced* if any proper convex subset of it has smaller minimal width. But the family of reduced bodies forms a proper superset of that of complete bodies (see below). It turns out that this is no longer the case in Minkowski spaces, see [797], [180], and our Chapter 10. Groemer [456] generalized the concept of reducedness calling a convex body \(K \subset {\mathbb E}^n\) *f*-*minimal* if, when *f* denotes an extended real-valued function on the family of convex bodies in \({\mathbb E}^n\), any convex body \(K'\) properly contained in *K* satisfies \(f(K') < f(K)\). In the cases when *f* denotes circumradius and inradius, all *f*-minimal bodies are determined in [456], and for smooth convex bodies and *f* standing for minimal width, *f*-minimality is shown to be equivalent to constant width. Other ways of generalization are given by leaving the concept of convex bodies, but staying with minimal width. For example, in [2] a connected arc of minimal width 1, say, is called reduced if any connected subarc of it has smaller minimal width. The authors want to determine, among all rectifiable planar arcs of minimum width 1, that arc having the least length. For example, the union of two adjacent sides of an equilateral triangle of altitude 1 has minimal width 1 and length 2.309...; it is proved in [2] that the required minimum length is approximately 2.2783, achieved by a caliper-shaped arc described in the paper. The relation to Buffon’s needle problem is clarified, and the authors ask for the shortest connected union of arcs of minimum width 1, also in higher dimensions.

We will survey now results on reduced bodies in Euclidean space. A wider representation is given in [701]. Since in \({\mathbb E}^n\) any constant width body is reduced and there are reduced sets not being of constant width, the superset statement should be shown by some examples of reduced sets which are not of constant width (see also [677] and [701], § 1), starting with the planar case. For example, every regular odd-gon is reduced, and there are also non-regular reduced odd-gons with at least five vertices. Further on, if we take, for polar coordinates \(\phi \) and *r* and each fixed \(k \in [\frac{1}{2}, 1]\), the convex hull of points such that \(|\varphi | \le (\pi -\mathrm{arc} \cos \frac{1-k}{k})/2\) and \(r \in \{1-k, k\}\), we get a planar reduced body. In particular, for \(k = \frac{1}{2}\) and \(k=1\) we obtain the unit disk and the quarter of a disk as extreme examples. Both these examples generalize also to \({\mathbb E}^n\), the latter as intersection of an *n*-ball, centered at the origin, and a closed orthant. If a planar reduced body has an axis of symmetry, we can rotate it about this axis to get a 3-dimensional reduced body of revolution, and this rotational procedure can be suitably generalized for the step from \({\mathbb E}^{n-1}\) to \({\mathbb E}^n\), see [701], § 2. The only centrally symmetric reduced bodies are the balls, see Claim 2 of [700], Remark 5.3 of [58], and also Theorem 7.4.2. A surprising result was obtained by Lassak [693]: Different to the planar situation, there are reduced bodies in \({\mathbb E}^n, n>2\) having fixed minimal width, but arbitrarily large finite diameter! It turns out that the family of reduced bodies in \({\mathbb E}^n\) has an interesting richness of forms, inspiring *geometric* research on it!

Regarding applications (e.g., various types of extremal and containment problems related to the notion of thickness (= minimal width); see, e.g., [46] and [451]), the following fact, immediately following from Zorn’s lemma, is important: Every convex body of minimal width *h* contains a reduced body of the same minimal width *h*. By the following, still open problem an attractive example is shown (we repeat that this problem inspired Heil [524] to introduce the notion of reducedness, see also the discussion A 18 in [272]): *Which convex bodies of fixed minimal width in* \({\mathbb E}^n, n > 2,\) *have minimal volume*? In the spirit of our “dualization remark” above, this can be seen somehow as a *dualization of the isodiametric problem* which, in Euclidean like also Minkowski spaces, has the ball as solution (see [820]). (It asks for the point set of fixed diameter that has the largest volume.) The solution to Heil’s question in \({\mathbb E}^2\) is the regular triangle, see [910]. And for higher dimensions this problem was already discussed in [160], Section 44, and [481], pp. 260–261; a good candidate for \(n=3\) is discussed in [524] and [481], and it is depicted in [524] and [143], Fig. 2.13. Oudet [906] introduced new numerical methods to solve optimization problems for convex bodies involving also minimum width as geometrical constraint. Besides discussing the possible optimality of Meissner’s bodies, the developed algorithms are used to approximate the optimal solution that Heil proposed for \(n = 3\). Further related investigations and results are presented in [356], [456], [209], [53], and [56], see also the discussion in [272, Problem A 18]. For example, in [209] it is shown that among all bodies of revolution in \({\mathbb E}^3\) the compact cone obtained as rotated equilateral triangle has, for fixed minimal width, minimal volume. We remind the reader here that even the restriction of this problem to bodies of constant width in \({\mathbb E}^n, n > 2\), is still open (with the Meissner bodies as conjectured extremal bodies), see also Section 8.3. The restriction to rotational bodies of constant width yields the rotated Reuleaux triangle as the only extremal set for \(n=3\), see again [209] and also Theorems 14.2.2 and 14.2.3. Replacing volume by quermassintegrals \(W_i, i=0,1,..., n-1\) (see [1039]), one can generalize the “dual isodiametric problem” staying within our framework, since the extremal bodies still have to be reduced (see [701], § 7]). For \(i = 1\) we get the original problem, and for \(i = n-1\) the solution is trivially given by the family of constant width bodies. For \(i = 1\) it was shown in [637] that for \(n > 2\) the balls are the only convex bodies of given minimal width having smallest surface area.

It would be essential to get a complete geometric picture how the boundary of a reduced body in \({\mathbb E}^n\) has to look like, e.g., also regarding symmetry conditions. Unfortunately, we are far away from this (except for the case \(n=2\)). Helpful statements, which are proved in this direction, are the following ones: Through every extreme point of a reduced body *R* in \({\mathbb E}^n\), one support hyperplane of a thickness strip passes, and every extreme point of *R* is one endpoint of a thickness chord of *R*. From these and many related observations (especially in [677] many of them are derived) one can get results of the following type: Every smooth reduced body in \({\mathbb E}^n\) is of constant width (claimed in [524] and proved by Groemer [456]), and every strictly convex reduced body in the plane is also of constant width (cf. [282]), see also Theorem 7.4.4. *It is not known whether for* \(n > 2\) *every strictly convex, reduced body in* \({\mathbb E}^n\) *is of constant width* (unfortunately, the approach of Nikonorov [893] is erroneous). Let *H* be a support hyperplane through a boundary point *x* of a convex body *K* in \({\mathbb E}^n\), and let *B* be a ball which is tangent to *H* at *x*; let *N* be a neighborhood of *x* such that the intersection of *N* and *K* belongs to *B*. Then *B* is said to support *K*, and if *K* is supported in this way almost everywhere in its boundary, it is called *almost spherically convex*. Dekster [283] proved that any reduced, almost spherically convex body in \({\mathbb E}^n\) is of constant width, see Theorem 7.4.6.

We come now to results on reduced bodies in \({\mathbb E}^2\). A very detailed study of their boundary properties is given in [677], see also [701], § 3. We call the union of two opposite sides of a nondegenerate convex quadrangle *Q* and the diagonals of *Q* a *butterfly* *BF*, and these two opposite sides the arms of *BF*. Based on this notion, in [677] (see also [328]) the following was established: the boundary of any reduced body *R* of minimal width *h* in the Euclidean plane consists of at most countably many pairs of “opposite” segments (namely occurring as pairs of arms of one butterfly, in each case) and of at most countably many pairs of opposite pieces of curves of constant width *h*. Thus *R* is in fact the convex hull of the endpoints of all its thickness chords. This inspired the following generalization of Blaschke’s result (see [131]) on the approximation of planar constant width bodies by constant width bodies whose boundaries consist only of circular arcs. Namely, Lassak [697] showed that for any planar reduced body *R* of minimal width *h* and an arbitrary \(\varepsilon > 0\) there exists a reduced body \(R'\) whose boundary consists only of circular arcs of radius *h* and of arms of butterflies of *R*, such that the Hausdorff distance between *R* and \(R'\) is at most \(\varepsilon \). Now we will discuss some metrical problems yielding also extremal reduced bodies. Since we are in a superset of the family of constant width sets, extremal sets from there (like the Reuleaux triangle) need no longer be extremal. In the following, we write *d*, *A*, and *p* for the *diameter*, *area*, and *perimeter* of a reduced body *R* of minimal width *h*. Already in [677] it was shown that the two inequalities \(d/h \le \sqrt{2}\) and \(p/h \le 2 + \frac{1}{2} \pi \) hold, in both cases with equality only for the quarter of a disk. On the other hand, the first ratio has the sharp lower bound 1 satisfied only if *R* is of constant width, and the second ratio has the sharp lower bound \(\pi \) with equality only if *R* is a disk. Due to the surprising result on arbitrarily large finite diameters in \(\mathbb {E}^n, n > 2\), derived in [697], there are no higher dimensional analogues of these results. Since the regular triangle is reduced, by [910] it follows that the area of a reduced body *R* of minimal width *h* satisfies \(\frac{1}{3}\sqrt{3} \cdot h^2 \le A\), and conversely in [695] the upper bound \(h^2\) for *A* was confirmed. A sharpening for reduced polygons is also given in [695], yielding the interesting problem whether each planar reduced body *R* of minimal width *h* satisfies \(A \le \frac{1}{4} \pi h^2\), with equality only for *R* a disk and a quarter of a disk. By Jung’s Theorem, any body of constant width *h* in the plane is contained in a disk of radius \(\frac{1}{3}\sqrt{3}h\). Lassak [697] showed that any planar reduced body *R* of minimal width *h* is contained in a disk of radius \(\frac{1}{2}\sqrt{2}h\), which is sharp for a quarter of a disk. It is well known that for each boundary point *x* of a set \(\Phi \) of constant width *h* the disk of radius *h* and centered at *x* covers \(\Phi \). Clearly, this does not hold for reduced sets in \({\mathbb E}^2\), but one can ask for special positions of disks doing this. This yields the open question *whether any planar reduced body* *R* *of minimal width* *h* *can be covered by a disk of radius* *h* *whose center is from the boundary of* *R*. For reduced polygons this is confirmed in [327]. On the other hand, since a related result of Blaschke [128] refers to the regular triangle, we have from this that any planar reduced body of minimal width *h* contains a disk of radius *h* / 3, characterizing this figure. Results on annuli formed by two concentric disks and containing the boundary of planar reduced sets are derived in [697].

Now we will discuss the geometry of reduced polygons. Their vertex number is odd, and under this supposition a polygon *P* is reduced with minimal width *h* if the orthogonal projection *y* of any vertex *x* onto the affine hull of the opposite side is an interior point of that side such that the distance between *x* and *y* equals *h*; moreover, *x* and *y* always halve the perimeter of *P* (cf. [677]). For reduced polygons *P* also the inequalities \(d/h \le \frac{2}{3}\sqrt{3}\) and \(p/h \le 2 \sqrt{3}\) hold, both sharp only for the regular triangle. This implies that every reduced polygon is contained in a disk of radius \(\frac{2}{3}\sqrt{3}\cdot h\), sharp only for the regular triangle, and it yields also (see again [677]) that among all reduced *m*-gons of minimal width *h* and with *m* not larger than *n*, only the regular *n*-gon has minimal diameter and minimal perimeter. We note that the statement on the perimeter was extended by [46] to arbitrary convex polygons. For the area of reduced polygons the inequality \(A < \frac{1}{4} \pi \cdot h^2\) holds, which in general cannot be improved.

We continue with results on reduced *n*-polytopes, \(n>2\). The *intriguing question whether there exist reduced* *n*-*polytopes in* \({\mathbb E}^n, n>2\), was posed in [677], see also [690]. It was shown in [792] that there are no reduced *n*-simplices (see [796] for \(n=3\), and for a large class of *n*-pyramids [61]). This is connected with the paper [128], where Blaschke erroneously assumed that the minimal width of a regular *n*-simplex is attained in the normal directions of its facets. (This statement, true only for \(n=2\), was corrected by Steinhagen [1090].) A deeper study of this problem was done in [60], where generalized antipodality notions were introduced as tools, and also the extension to Minkowski spaces is considered. For the Euclidean subcase it was shown there that any *n*-dimensional polytope (\(n>2\)) having *r* facets and *s* vertices is not reduced in the cases \(r = n+2\), \(s = n+2\), and \(s > r\) (therefore, no simple *n*-polytope is reduced). And in [700] centrally symmetric polytopes were excluded. After proving a new necessary condition for *n*-polytopes, \(n>2\), to be reduced, the authors of [437] surprisingly succeeded in constructing a 3-dimensional Euclidean reduced polytope!