Complete sets in normed linear spaces

A bounded subset of a (finite or infinite dimensional) normed linear space is said to be complete (or diametrically complete) if it cannot be enlarged without increasing its diameter. Any bounded subset A of a normed linear space is contained in a complete set having the same diameter, which is called a completion of A. We survey characterizations, basic properties, facts about structure of the interior and boundary, and the asymmetry of complete sets. Different methods to obtain completions of bounded sets are presented. Moreover, the structure of the space of complete sets endowed with the Hausdorff metric and relations of this set family to related set families and notions are discussed. For example, we mean here sets of constant width, balls, reduced sets, sets of constant diameter, and sets of constant radius.


Introduction
There are three important and well known types of convex bodies in classical convexity that also play an essential role in real Banach spaces and are strongly related to each other: bodies of constant width, diametrically complete sets, and reduced bodies. Basic references collecting the known material about two of these three types are: for bodies of constant width §15 of the classical monograph [16] (or [17] as English translation), the surveys [30,49,93], as well as the whole book [88]; and treating reduced bodies the two surveys [76,77]; see also Chapters 7 and 10 of the monograph [88]. A comprehensive and survey-like publication on diametrically complete sets in Euclidean and real Banach spaces is still missing, and we hope to fill this gap with the present exposition.
We continue with some historical remarks. It seems that the notion of complete set goes back to Meissner [97] who proved that in the Euclidean plane and in Euclidean 3-space each complete set is of constant width (the converse is clear). Jessen [55] extended this result to higher dimensions, and Reidemeister [126] characterized, related to this, the bodies of constant width as those of constant diameter. Pál [114] (for dimension n = 2 ) and Lebesgue [78] proved that each set A, in a Euclidean space, of diameter diam A is contained in a complete set C of diameter diam A , and therefore in a set of constant width diam A . The set C is then called a completion of A, and in general C is not uniquely determined. Also for diametrically complete sets we refer to [16] (or [17]), namely to Section 64 there. This is the first monograph in which complete sets got an own section where they are explicitly discussed. After that book, results on complete sets were collected only in overviews treating sets of constant with; see Section 4 of [30], Subsection 5.4 of [49], Subsection 2.5 of [93], as well as Chapter 7, Sections 10.3 and Section 10.4 (with the corresponding Notes, see pp. 231-235 and pp. 241-242) of the monograph [88]. In addition, Chapter V of the book [14] contains several related results. And in Section 2.5 (dedicated to bodies of constant width) of the monograph [140] also diametrically complete sets are discussed. E.g., related notions like wide spherical hull, tight spherical hull and diametric completion map occur there.
We say that y ∈ A is a nearest point to x in A if The number (x, P) is called the extreme porosity of P at x. If (x, P) > 0, ∀x ∈ P , then we say that P is very porous (or lower porous). If (P) > 0 , then we say that P is uniformly very porous. Let X = (X, ‖⋅‖) be a real normed linear space with origin o , unit ball B X , and unit sphere S X . X is said to be strictly convex if S X does not contain a nontrivial segment; it is smooth if there exists a unique supporting hyperplane of B X at each point in S X . For each point x ∈ X and each > 0 , we set B X (x, ) ∶= x + B X . This means that B X (x, ) is the ball centered at x and having radius . If X is finite dimensional, (X, ‖⋅‖) is called a Minkowski space. A two-dimensional Minkowski space is called a Minkowski plane. A Minkowski space whose norm is induced by an inner product is referred to as a Euclidean space. An n-dimensional Minkowski space is always isometric to a Minkowski space on ℝ n . Thus, n-dimensional Minkowski spaces, here under consideration, are assumed to have the form (ℝ n , ‖⋅‖).
Let A ⊆ X . The convex hull and closed convex hull of A are denoted by conv A and clconv A , respectively. For two distinct points x and y in X , the segment between these two points, the line passing through them, and the ray starting from x and passing through y are denoted by x, y , ⟨x, y⟩ , and � x, y⟩ , respectively. When x = y , we set x, y = {x} and say that the segment is trivial.
We denote by d ‖⋅‖ (⋅, ⋅) the metric on X induced by ‖⋅‖ . Thus If A ⊆ X is bounded and nontrivial, then and Moreover, the functional (A, ⋅) ∶ X → ℝ is convex. When A is a compact convex set in a Minkowski space X , we have

4) diam A ≥ (A, A) + � (A) ≥ (A) + � (A).
A boundary point x of a bounded, closed and convex set K in a normed linear space is called maximally diametral if it is diametral and not properly contained in a diametral set with the same diam-If X is infinite dimensional, then J s (X) ≥ √ 2 . When X is a nonreflexive Banach space, J s (X) = 2 (see [4,82]). When J s (X) < 2 , X is said to have the uniform normal structure. Note that the conditions "X has normal structure" and " J(X) < 2 " are independent, cf., e.g., [103, p. 14].
For each number ∈ (0, 2] , the modulus of convexity X ( ) is defined by The characteristic of convexity 0 (X) is defined by A space X is said to be uniformly convex if 0 (X) = 0.
Let (X, ‖⋅‖) be a Banach space. It is said to be locally uniformly rotund (LUR) if {x n } ∞ n=1 tends strongly to x whenever x ∈ X and {x n } ∞ n=1 satisfies We say that (X, ‖⋅‖) has the Kadec-Klee property with respect to the weak topology (the Kadec-Klee property for short) if each sequence {x n } ∞ n=1 with lim n→∞ ‖ ‖ x n ‖ ‖ = 1 and converging weakly to a point x ∈ S X tends strongly to x. This space is said to have the non-strict Opial property if, for any sequence {x n } ∞ n=1 weakly converging to a point x and each point y ∈ X , we have A compact convex set having interior points in ℝ n is called a convex body. The collection of convex bodies in ℝ n is denoted by K n , and the set of elements in K n that are symmetric with respect to o is denoted by C n . It is clear that K ⊆ ℝ n is the unit ball of a Minkowski space on ℝ n if and only if K ∈ C n . In view of this, when K ∈ C n is the unit ball of a Minkowski space having some property P, we say that K has the property P, and vice versa.
Let K and L be two compact convex sets with interior points in a Minkowski space X. The Banach-Mazur distance d BM (K, L) between them is defined by where A is the set of nonsingular affine transformations from X to X.
Let Ω be a compact Hausdorff space, and C(Ω) be the Banach space of continuous real functions on Ω with the supremum norm ‖⋅‖ ∞ . Let f ∶ Ω → ℝ be bounded. For each accumulation point x of Ω , set If x is an isolated point of Ω , then we set Euclidean width if and only if adding any point exterior to it increases the diameter with respect to the Euclidean metric. There are two different ways to extend the notion of sets of constant Euclidean width to normed linear spaces. The first one uses "maximal with respect to the diameter" as the characteristic property.

Definition 1
Let K ⊆ X be bounded. If then we say that K is diametrically maximal, diametrically complete, or simply complete.
We denote by D the collection of all complete subsets in X. The other way is more "natural". Let K be a bounded closed convex subset of a normed linear space X . For each f ∈ S X * , where S X * is the unit sphere of the dual space X * of X, the number is called the width of K in the direction of f. Clearly, for each f ∈ S X * we have Note that (see, e.g., [29,Proposition 4]) The number is called the minimal width of K. When a normed linear space X is subreflexive (i.e., when the set of norm attaining functionals is dense in X * ), we have then K is called a set of constant width.
In [36] and §2 of the dissertation [39] it is explicitly proved that in normed planes completeness and constant width are equivalent notions, and that in higher dimensions constant width still implies completeness. In fact, it can be proved by the Hahn-Banach Separation Theorem (see e.g., [96,Proposition 1.9.15]) that the following theorem holds.
Remark 4 A complete set in X is not necessarily of constant width. For example, let X = (ℝ 3 , ‖⋅‖ 1 ) and Then, A is a simplex circumscribed about B X , which is an octahedron. It can be verified that A is a complete set whose diameter is 4 (see Fig. 1). Clearly, A is not of constant width ( Δ(A) = 2 < diam A ) and not a ball. This example negates the main result in [66].
Moreno and Schneider [106] wrote that "the class of bodies of constant width, which has such a rich theory in Euclidean spaces, is rather poor in typical normed spaces, and that the class of complete sets should be studied instead." Papini collected in [115] some results on the role that completeness plays in approximation theory. Stefani [138] studied measure-theoretical problems and approximation aspects related to completeness and constant width (see also further papers of him cited in [138] and [139]). As a more general concept, Groemer [44] studied socalled f-maximal and f-minimal sets, with applications in different fields (geometric inequalities in convexity, or packing and covering constants in the geometry of numbers). Here f stands for various functionals of convex bodies, such as inradius, circumradius, minimal width, diameter and so on. The latter subcase (where f denotes the diameter) yields the class of complete sets as such extremal sets. The notion of complete sets (in Euclidean spaces) was recently rediscovered in [51].

Elementary properties and characterizations
The definition of complete sets shows that 1. ∅ is not complete while a singleton is always complete; 2. A complete set is bounded, closed, and convex; 3. A closed ball in X is complete; 4. If K is complete and ∈ ℝ , then K is also complete.
As mentioned in Remark 4, a complete set is not necessarily a ball. Moreover, we have the following characterization of closed balls among the collection of complete sets. Let K be a bounded closed convex set. For each f ∈ S X * , put Each of the sets H + f and H − f is a supporting hyperplane of K. The set S f is called the supporting slab of K determined by f, whose width is w f . A boundary point x of K is said to be smooth if there exists a unique supporting hyperplane of K passing through x. If H + f ∪ H − f contains a smooth boundary point of K, then S f is said to be regular. If S f is parallel to a regular supporting slab of B X , then we say that S f is B X -regular. Let a, b ∈ K . If there exists f ∈ S X * such that a ∈ H + f and b ∈ H − f , then [a, b] is called an affine diameter (or a diametrical chord) of K generated by S f ; for properties of affine diameters we refer to Soltan's excellent survey [135]. If the supporting slab generating an affine diameter [a, b] of K is regular, then [a, b] is called a regular affine diameter. It is shown in [131] that if K is a compact convex set having interior points in a Minkowski space X , then K is the intersection of all its regular supporting slabs. Based on this result, Moreno and Schneider proved the following characterization of complete sets in Minkowski spaces.
Complete sets in C(Ω) have the following characterization.

Theorem 8 [110] A nontrivial set K ⊆ C(Ω) is complete if and only if
1. K has the form [f, g], 2. there exists a dense subset S of Ω , such that f and g are both continuous at each 4. either f is lower semicontinuous and g is upper semicontinuous, or f is strongly upper semicontinuous and g is strongly lower semicontinuous.
Theorem 9 (cf. [36]) Let Y = (ℝ m , ‖⋅‖ 1 ) and Z = (ℝ n , ‖⋅‖ 2 ) be two Minkowski spaces, and X = (ℝ m+n , ‖⋅‖) be the Minkowski space whose unit ball B X is the cartesian product of B Y and B Z . If K ⊆ X is complete, then it is the cartesian product of K 1 ⊆ Y and K 2 ⊆ Z , which are complete in Y and Z, respectively.
When the underlying space is finite dimensional, we have the following simple characterization of sets of constant width.

Proposition 10 [36] A compact convex set K in a Minkowski space is of constant width if and only if K − K , the Minkowski sum of K and −K , is a ball.
A general characterization of sets of constant width in Banach spaces is due to Payá and Rodríguez-Palacios.
Theorem 11 [120] Let K be a nontrivial bounded closed convex set in a Banach space X . The following conditions are equivalent: As noted in [104], if K is of constant width in a reflexive Banach space or in the dual space of some Banach space, then K − K is a ball.
In the rest of this subsection, we summarize some known characterizations of planar sets of constant width.
Averkov and Martini [5] proved that a convex set K in a Minkowski plane is of constant width if and only if every chord of K splits K into two compact sets such that one of them has diameter equal to the length of this chord.
If p 1 , p 2 , p 3 are three distinct points in a Minkowski plane X satisfying then the set is a set of constant width ‖ ‖ p 1 − p 2 ‖ ‖ , and is called a Reuleaux triangle. Martini and Mustafaev [89] proved that there exists a Reuleaux triangle in a Minkowski plane homothetic to the unit disc if and only if the unit circle is either an affine regular hexagon or a parallelogram. See Fig. 2, where ∞−1 is the Minkowski plane on ℝ 2 with the following norm: The ratio of the area of the unit disc of a Minkowski plane to that of a Reuleaux triangle having diameter 1 is between 4 and 6. They also extended Barbier's theorem to Minkowski planes by showing that the perimeter of a convex set of constant width in a Minkowski plane X equals ∕2 times the perimeter of B X . Let X be a normed linear space and A ⊆ X . The image of A under an isometry on X is called a congruent copy of A. Spirova [137] proved that if every congruent copy of a compact convex set K in a Minkowski plane can be covered by a translate of any Reuleaux triangle of diameter , then each congruent copy of K can be covered by a translate of any convex body of constant width .
Let x and y be two elements in a normed linear space X. If then we say that x is Birkhoff orthogonal to y and we write x ⟂ B y (cf., e.g., [1] or [2] for more information on this orthogonality type). Let K be a planar convex body. is said to be a normal of K at p if p − q is Birkhoff orthogonal to the direction of a supporting line of K at p. Let [p 1 , p 2 ] and [q 1 , q 2 ] be two chords of K, and L be the If either p 1 + L and p 2 + L both support K, or p 1 + L intersects bd K in precisely two points p 1 and p ′ 1 and [p � 1 , p 2 ] is an affine diameter of K, then we say that [p 1 , p 2 ] is affinely orthogonal to [q 1 , q 2 ] through p 1 , and we write Proposition 12 [3,Theorem 4.6] Let K be a convex body in a strictly convex and smooth Minkowski plane. The following assertions are equivalent: Let [q 1 , q 2 ] be a chord of a planar convex body K, and p ∈ bd K . If there exists an affine diameter

Completions of bounded sets
Definition 14 Let A be a bounded set in a normed linear space X . If A C is a complete set containing A and diam A C = diam A , then A C is called a completion of A.
By Zorn's Lemma, each bounded set A in a normed linear space X has at least one completion. When X is a Minkowski space, a completion of A can always be constructed, see Sect. 4. For each bounded subset K of a normed linear space X , we denote by C(K) the set of all completions of K. Therefore, C(⋅) is a set valued map defined on the set of bounded subsets of X , and it will be called the diametric completion map.
Groemer [45] showed that if X is a Minkowski space, then for each K ∈ H there is a completion of K with maximal volume. If X is strictly convex, such a completion is unique and defines a map called the maximal volume completion map.
Let K c ∈ C(K) . If K and K c have a common circumball, then K c is called a Scott completion of K. In a Minkowski space, a bounded closed convex set K always has a Scott completion (cf. [134,141], and [20, Corollary 2.7]).
Sallee [128] introduced a way to obtain, in infinitely many steps, a complete set in n . A similar but more general construction can be found in [130].
For a nontrivial subset K of C(Ω) , one can give a completion of K explicitly.
Let K and L be two distinct complete sets in a Banach space X. If there is no inclusion between them, then K ∪ L is not complete; . Therefore, neither the union nor the intersection of two distinct completions of a bounded set is complete. One can also verify (see [119]) that or, equivalently,

For each bounded set A, the intersection of all balls containing A is called the ball hull of A, denoted by B(A).
In C(Ω) , we have the following characterizations of ball hull, wide spherical hull, and tight spherical hull.

Moreover, if one of W(A) and T(A) is of constant width, then W(A) = T(A)
. These properties show that diametrical completeness is closely related to the theory of ball hull and ball convexity, cf. [11,54,69,92]. One can also verify that (cf. [48]) Let A and B be two sets in a Banach space X. It is shown in [90, Theorem 2 and Proposition 3] that

then W(A) ≠ W(B) and T(A) ≠ T(B); 2. W(A) = W(B) if and only if T(A) = T(B).
Concerning the structure of bd A , bd (T(A)) , and bd (W(A)) , we have the following result.

A point x ∈ A is a diametral point of A if and only if it is a diametral point of T(A).
Proof 1. In [117] it is shown that However, a contradiction. Thus x ∈ bd (W(A)).
2. First suppose that x ∈ bd (T(A)) . Suppose to the contrary that

Proposition 18 [48] For each A ∈ H , we have
When the underlying space is finite dimensional, then both A and T(A) have strongly diametral points. This is not true in general: both T(A) and A can have no diametral point, and it is possible that Example 3 [48] We denote by X the Banach space (c 0 , ‖⋅‖ ∞ ) , where ‖⋅‖ ∞ is the usual maximum norm, and by Y the Banach space having this property will be called a non-increasing permutation of . It is not difficult to verify (see, e.g., [147, p. 1774 The space Y is locally uniformly convex and the norm ‖⋅‖ D is equivalent to ‖⋅‖ ∞ on c 0 (see [125]). Therefore, if In general, the wide spherical hull of a bounded set A may have empty interior, see Sect. 7 below. However, this cannot happen when ri A ≠ ∅ , where ri A is the relative interior of A with respect to the affine hull of A.
Proof Let x be an arbitrary point in ri A . Then, there exists a positive number such that B X (x, ) ∩ aff A ⊆ A . Let y be an arbitrary point in A. Then In the remaining part of this subsection, we collect some results concerning the continuity of the wide spherical hull map and the tight spherical hull map.
For two compact convex subsets K and L in a Minkowski space X , put .
If, for each bounded subset K of a normed linear space X, [100] showed that the diametric completion map C(⋅) is generally not convex valued. For example, it is not convex valued in 3 1 . Based on characterizations of complete sets in C(Ω) , Moreno [101] proved that every K ⊆ C(Ω) that can be expressed as the intersection of closed balls has a nonempty metric projection (in terms of the Hausdorff metric) in C(K) as well as a nonempty metric projection in the collection of all complete sets. He also characterized the case when the metric projection of K in C(K) is a singleton.
Recall that a topological space is said to be extremally disconnected if every open set in this space has an open closure.

Sets having a unique completions
In general a bounded set may have more than one completion. It is clear that a closed segment of length 1 in the Euclidean plane ℝ 2 has different completions. See [68, Corollary 3.2] for a characterization of convex bodies having a unique completion in the Euclidean plane. Papini proved that any Banach space X with dim X ≥ 2 contains a segment having different completions, see [117,Theorem 3.7]. Groemer (see [45]) proved that sets having a unique completion are nowhere dense in a strictly convex Minkowski space. This result is improved by the following A Banach space X is said to be uniformly nonsquare if 0 (X) < 2 . Papini [117,Theorem 3.7] proved that in a uniformly nonsquare Banach space every segment has different completions, and that, as already mentioned above, in any Banach space whose dimension is at least 2, there are segments having different completions. Therefore, it is natural to consider the following property: (U 1 ) There exists a segment S ⊆ X having a unique completion.
Clearly, if a nontrivial segment [a, b] has a unique completion C, then C is the ball centered at (a + b)∕2 and having radius ‖a − b‖∕2.
Theorem 33 [47,Theorem 3] Let u ∈ S X . Then the following facts are equivalent: 1. [−u, u] has a unique completion, 2. the unit circle S L of each two-dimensional subspace L of X containing u is a parallelogram having u as one of its vertices.
Therefore, a normed linear space X has property ( U 1 ) if and only if there exists u ∈ S X such that the unit circle S L of each two-dimensional subspace L of X containing u is a parallelogram having u as one of its vertices. We say that X has property (U b m ) if there exists an m-dimensional equilateral simplex in X whose unique completion is a ball. And we say that X has property (U m ) if there exists an m-dimensional equilateral simplex in X having a unique completion. Clearly, a space having property ( U b m ) has also property ( U m ). The Euclidean plane has property ( U 2 ) but does not have property ( U b 2 ).
Proposition 34 [47] Let X be a normed linear space whose dimension is at least 2. If X has property ( U 1 ), then it has also properties ( U b 2 ) and ( U 2 ).
We do not know whether ( U 2 ) implies ( U 3 ). In general, one may consider whether the implication ( U m )⟹(U m+1 ) is true for each m ∈ ℤ + .
The following result characterizes sets in C(Ω) having a unique completion.
Theorem 35 [110] Let K be a nontrivial subset of C(Ω) . The following conditions are equivalent: 1. K has a unique completion, 2. B(K) is complete, Let A be a nontrivial bounded set in a Minkowski space. The smallest positive integer m such that A can be represented as union of m sets with diameters less than diam A is called the Borsuk number of K in X, denoted by b X (A) . Boltyanski [13] proved that, in the Euclidean plane, the Borsuk number of a nontrivial bounded set K is 3 if K has a unique completion. In general Minkowski planes, it is known that if a bounded set K has different completions then b X (K) = 2 (see, e.g., [14,Lemma 33.2]). In [15], Boltyanski and Soltan proved that, if a compact set K in a Minkowski plane has a unique completion K c and, for each pair of parallel supporting lines of K c , at least one of them intersects K, then b X (K) ≥ 3 (see also [14,Theorem 33.9] or [71, Theorem 1.1] for a summary of results in this direction). Martini et al. [91,Proposition 4] proved that if K is a nontrivial set of constant width in a Minkowski plane X, then b X (K) = 3 if B X is not a parallelogram; b X (K) = 4 otherwise. Generally, if K is a nontrivial set of constant width in an n-dimensional Minkowski space X, then b X (K) ≥ b X (B X ) ≥ n + 1 (cf. [91,Theorem 9]). Following Zong's quantitative method for attacking Borsuk's conjecture in Euclidean spaces (cf. [148]), Lian and Wu [79] proved that, for each p ∈ [1, ∞] , any set in 3 p having diameter 1 can be expressed as the union of 8 sets whose diameters are at most 0.925. Zhang et al. [146] improved this result by replacing 0.925 with 0.9. Lángi and Naszódi [71] studied whether a so called k-fold Borsuk covering of a closed set having a unique completion extends to a k-fold Borsuk covering of its unique completion when the underlying space is two-dimensional. It is shown that the k-fold Borsuk number of a bounded set K, which has a unique completion K c distinct from a Reuleaux polygon such that each pair of parallel supporting lines l and l ′ of K c satisfies (l ∪ l � ) ∩ K ≠ � , equals 2k + 1.

Eggleston's construction
In this subsection we present a completion method (cf. [119]) introduced by Eggleston [35] for Euclidean spaces, which is also valid in Minkowski spaces. Let K be a compact subset of a Minkowski space X . Put Then K 1 is compact and diam K 1 = diam K . In general, let K i be already defined for some i ≥ 1 . Then K i is a compact set satisfying If K i is complete, then it is a completion of K, and this process is done. Otherwise, We would obtain a strictly increasing sequence {K i } ∞ i=1 unless the process stops for some finite i. Let If K c is not complete, then there exists a point such that For each i ≥ 1 and each j ≤ i we have Therefore, B X (y, diam (W(K))) contains an infinite -separated sequence, a situation that never happens in a Minkowski space. Thus K c is complete. Papini and Wu [119] provided the following concrete example showing that Eggleston's construction does not apply to infinite dimensional spaces, even when the space is separable.

Example 4 Let X = C([−1, 1]) and
Then Consider Eggleston's construction by taking f k ∈ X such that, for each integer k ≥ 1, where Then, for each function f ∈ K c , Therefore K c is not complete.
Theorem 36 provides a construction similar to Eggleston's, which works in separable Banach spaces for sets in H having interior points.
Theorem 36 (cf. [119]) Let X be a separable Banach space, {x i |i ∈ ℤ + } be a fixed dense subset of S X , and K 0 ∶= K be a set in H containing o in its interior. For each i ≥ 1 , let x i be the intersection point of � o, x i ⟩ and bd (W K i−1 ) , and let is a completion of K.

The Maehara set of a bounded set
Let K be a bounded subset in a normed linear space. The set is called the Maehara set of K. In general, M(K) is not a complete set. See, e.g., [109, Fig. 1]. Maehara [80] showed that M(K) is a completion of K in Euclidean spaces. This result was extended by Sallee [129] to Minkowski spaces X satisfying the condition that W(A) is a summand of B X whenever A ⊆ X has diameter 1. Polovinkin [122] showed that this result also holds in reflexive Banach spaces with generating unit ball, where a set K is called a generating set if any nonempty intersection of translates of K is a summand of K. Topological and algebraic properties of generating sets and related notions (like M-strongly convex hulls) were deeply studied in [6]. In a Banach space with a generating unit ball, every closed halfspace defined by a norm attaining functional and containing T(K) for some K ∈ H contains a completion of K (cf. [109,Lemma 3]). The collection of generating sets is closed under linear transformations and direct sums. Two-dimensional convex bodies are generating (see [41]), like also the balls of Hilbert spaces (cf. [6,80]). See [95] for more information on generating convex polytopes (called strongly monotypic polytopes there) and [19] for more examples of generating sets. And in [18] the investigations from [95] are continued: the class of all 3-dimensional polyhedral generating sets coincides with the family of all strongly monotypic 3-polytopes classified in [95]. Ivanov [53] found a nice property of generating sets yielding some new symmetric examples of them in Hilbert spaces. After giving a survey on related own results, Polovinkin [123] presented constructive approaches to completions of sets in reflexive Banach spaces having generating sets as unit balls. Using the machinery of strongly convex analysis developed in [121], he established conditions for unique completions to bodies of constant width, and algorithmical approaches to all bodies of constant width containing a given set of the same diameter, see also [124]. Interesting separation results, where generating sets play an essential role, are discussed in [50]. And also in Chapter 3 of Schneider's book [131] properties of generating sets and related results are comprehensively discussed and summarized. Moreno and Schneider [109] mentioned the problem of characterizing spaces X so that M(K) is a completion of K for each K ∈ H (the original problem is stated for Minkowski spaces). And they wrote: "We do not know whether the Minkowski spaces in which every Maehara set is complete (without necessarily being of constant width) have a simple characterization". This situation indicates that the structure of wide spherical hull and tight spherical hull of bounded sets is still not perfectly known.

Proposition 38 [106, Proposition 2] Let X be a Minkowski space, and A ⊆ X be a set having diameter 1. If one of the sets W(A) and T(A) is a summand of B X , then M(A) is a body of constant width 1 that contains A.
A normed linear space X is said to have the Karasëv property if, for each u ∈ B X , B X ∩ (B X + u) is a summand of B X .

Theorem 41 [109, Theorem 4] Let X be a Banach space with a generating unit ball and K ∈ H . Then, for each C ∈ C(K),
In Euclidean spaces, the Maehara completion of a convex body K is at least as smooth as K itself, see [81] and [109,Theorem 5].

Generalized Bückner completion
In this subsection, we assume that X is a Minkowski space. Moreno and Schneider [109] obtained a locally Lipschitz continuous selection of the diametric completion mapping by extending Bückner's completion procedure (cf. [23]) to Minkowski spaces.
For each L ∈ H and each u ∈ X ⧵ {o} , set Since X is finite dimensional, there exist finitely many vectors u 1 , … , u m ∈ X ⧵ {o} such that Let K ∈ H . It can be verified that the set has diameter diam K and inradius at least diam K 2(n+1) (see Section 4 in [109]). Therefore, by replacing K with 1 2 K + 1 2 W(K) and applying a suitable translation if necessary, we may assume that diam K 2(n+1) B X ⊆ K . It follows that Let It is proved in [109] that K m is complete, and it is called the generalized Bückner completion of K. Moreover, the generalized Bückner completion is a locally Lipschitz continuous selection of C(⋅) with a constant which depends only on the dimension of the space.

Completions related to hyperplanes
Bavaud [9] proved that, if X is the Euclidean plane, L is a line intersecting K ∈ H in a segment whose length is diam K , and L + and L − are the two closed halfplanes bounded by L, then are two completions of K. A more general and more complicated construction based on a similar idea was presented by Lachand-Robert and Oudet in [10]. Let X be a Banach space whose dimension is at least two, and H be a closed hyperplane of X containing the origin o . We denote by H + and H − the two closed halfspaces determined by H. Put, for a bounded subset K of X, and, moreover, Moreover, both K a and K b are convex (see [119,Proposition 3]). The following condition is important for the construction in this subsection.
The following construction is suggested in [9, p. 320] for K = K 0 and a line H in the Euclidean plane X. Let K be a set contained in the hyperplane H and K 1 a (not necessarily convex) set satisfying The set K C depends not only on K, but also on the starting set K 1 . Clearly, K C ∩ H contains K and has the same diameter. If K is complete and K C is a completion of K in X , then K C ∩ H = K. [119,Theorem 5] If the condition (dc) holds for K and if K 1 is a set satisfying (4.3), then the set K C obtained by the above construction (C) is a completion of K, as well as of K 1 .

Theorem 43
By interchanging the roles of H + and H − in the procedure above, we shall obtain (under the assumption that (dc) holds) another completion of K. If we take K 1 = K , then we shall denote K C by K A , and by K B the corresponding set when we reverse the roles of H + and H − . Papini and Wu proved that if K ⊆ H is a set satisfying (dc), then all completions of K can be obtained by the method of Theorem 43 (cf. [119,Theorem 8]).
Condition (dc) certainly holds if, for a hyperplane H , K ⊆ H has a unique completion K c in X. In fact, in this case, W(K) = T(K) = K c , so diam (W(K)) = diam (K) . The next result stresses the importance of (dc) for completions in this subsection. [119,Proposition 9] Given K ⊆ H , the following are equivalent:

Proposition 44
Theorem 45 [119,Theorem 12] Let K be contained in the hyperplane H, and K 1 be a set satisfying (4.3). Then K a , K b as well as K C are complete if one of the following conditions holds: 1. X is a strictly convex Banach space satisfying 0 (X) ≤ 1 and K is a set having constant width in H.

A stochastic construction
Moreno [102,Proposition 5] proved that, in a strictly convex Minkowski space, the following stochastic process yields a complete set C with probability one. Choose a positive and an arbitrary point x 1 . Put C 1 = {x 1 } . Let x 2 be a point uniformly chosen in B X (x 1 , ) , and put C 2 = conv {x 1 , x 2 } . Suppose that x i has been chosen for each i ≤ n and C n is fixed. Choose uniformly x n+1 from x ∈ X | | | | (C n , x) ≤ , and Then {C n } n∈ℤ + is a bounded and increasing sequence. The following set has the required property:

Completions to sets of constant width with preassigned shape
In this subsection, we focus on completions of bounded subsets in Euclidean spaces. In this situation, every completion of a nontrivial bounded subset is of constant width. For each x ∈ ℝ n and each > 0 , denote by B(x, ) the Euclidean ball centered at x having radius . Let x be a boundary point of a compact set K, Schulte [132] proved that, a compact convex subset C of ℝ n has a completion K of constant width such that each point in C ∩ bd K is an endpoint of a diametral chord of C, the symmetry group of C is contained in that of K, and that each singular boundary point x of K is a singular boundary point of C (relative to the affine hull of C) and satisfies S(x, K) = Sing (x, C). Let L be a subset of ℝ n of constant Euclidean width 2, be a positive number such that a Euclidean ball having radius can slide freely in L, be a number in (2 − , 2) , and C be a compact convex subset of L having diameter . Schulte and Vrećica [133] proved that there exists a completion K of C such that K ∩ bd L = C ∩ bd L , each point in (C ∩ bd K)⧵ bd L is an endpoint of a diametral chord of C, each common symmetry of C and L is a symmetry of K, and that each singular boundary point x of K is a singular boundary point of C (relative to the affine hull of C) and, if x ∈ K⧵ bd L , satisfies S(x, K) = Sing (x, C) . In particular, if C is a compact subset of bd L having diameter , then there exists a completion K of C such that K ∩ bd L = C , each common symmetry of C and L is a symmetry of K, and each singular boundary point x of K lies in C and is a singular boundary point of the convex hull of C (relative to the affine hull of C).

Relations to sets of constant width and to balls
We may consider the following properties that a normed linear space X might have: (A) every complete set in X is of constant width; (B) every complete set in X is a ball; (C) every convex body of constant width in X is a ball.
As we have mentioned in Sect. 3, sets of constant width are complete and the converse is not true. However, in many Banach spaces (e.g., Eggleston solved this problem for dim X = 2 by showing that each Minkowski plane is perfect (cf. [36]). He also showed that there are three-dimensional Minkowski spaces which are not perfect.

Proposition 48 A normed linear space X is perfect if and only if each nontrivial bounded set A is contained in a set of constant width diam A.
Proof First suppose that X is perfect. By Zorn's Lemma, A has a completion A C which is, by the assumption, a set of constant width diam A.
Conversely, let K be an arbitrary nontrivial complete set. The hypothesis shows that K is contained in a set K ′ of constant width diam K . Since the diameter of K ′ is diam K , K = K � , which shows that K is of constant width. Thus X is perfect. ◻ The finite dimensional situation of the following result can be traced back to Meissner. See, e.g., [143,Theorem 7.6.5] for a proof.  From results of Maehara [80] and Sallee [129], Polovinkin [122] derived that if the unit ball is a generating set, then the corresponding norm is perfect.
Following an idea of Maehara, in a Minkowski space X, a pair (K, L) of sets in H is said to be of constant width if K − L = B X . In [91], the following condition was characterized: (S2) (K, L) is of constant width if and only if Sallee [129] proved that, a Minkowski space whose unit ball is a polytope satisfies (S2) if and only if its unit ball is monotypic, where a centrally symmetric polytope P in a Minkowski space X is monotypic if and only if, for every x ∈ X , P ∩ (P + x) is either empty or a Minkowski summand of P, see [95].

Theorem 51 [91] A Minkowski space X satisfying (S2) is perfect.
Having the Karasëv property is sufficient but not necessary for a Minkowski space to be perfect (cf. cf. [65]). The next result provides necessary conditions for a Minkowski space to be perfect.

Proposition 52 [106, Proposition 3 and Theorem 4] Let X be a Minkowski space.
1. If X is strictly convex and perfect, then B X ∩ (B X + u) is a summand of B X whenever u ∈ B X . 2. If X is perfect, then 1 2 (B X ∩ (B X + u)) is a summand of B X whenever u ∈ B X .
Moreno and Schneider provided an example showing that, in general, the factor 1/2 cannot be replaced by a larger one. When the unit ball of the underlying space is a polytope, this factor can be chosen depending on the facial structure of the unit ball and could be larger than 1/2. More precisely, they proved the following Theorem 53 [106,Theorem 5] Let X be a perfect Minkowski space whose unit ball B X is a polytope, and be the maximal diameter of proper faces of B X . Then is a summand of B X whenever u ∈ B X .
It is known that a Minkowski space X has property (B) if and only if B X is a parallelotope, see [36,Corollary 2] for the sufficiency, [136] for the necessity, and [46] for the planar case. For the general case, there are similar characterizations. A metric space (X, d) is said to be hyperconvex if for any indexed family of closed balls {B X (x i , i )} i∈I of X satisfying the condition one always has A Banach space X is called a P 1 space if X is norm 1 complemented in every Banach space containing it.

Theorem 54
Let X be a real Banach space. The following facts are equivalent: The equivalence (1) ⟺ (2) is proved in [40,Theorem 2], and the equivalence (2) ⟺ (3) is obtained in [32], while the equivalence (3) ⟺ (4) is proved by Nachbin in [111] (the term "Banach space having the binary intersection property" is used there instead of hyperconvex). See also [112 In an n-dimensional Minkowski space having property (B), any bounded set having diameter 1 is contained in a closed ball having the same diameter. Such a ball is an n-dimensional parallelotope, and therefore it is the union of 2 n smaller balls whose diameters are all 1/2. This simple fact has been used in the study of Borsuk's partition problem in Minkowski spaces. See, e.g., [79,142]. For p ∈ (1, ∞) , Ivanov [52, Theorem 2.1] provided a sufficient condition for a set of constant width in n p to be expressed as the union of n + 1 subsets having smaller diameters.
In a Minkowski space X , a compact convex set A symmetric about the origin is reducible if there is a nonsymmetric closed convex set B such that A = B − B . If A is not reducible, it is irreducible. Yost [145] proved that, when n ≥ 3 , most (in the Baire category sense) n-dimensional Minkowski spaces have a smooth and strictly convex unit ball, have property (C), and hence do not have property (A) (since properties (C) and (A) together imply property (B)).
Let Ω be a compact Hausdorff space. It is shown in [104] that Recall that a pseudoball is a bounded closed convex set whose weak * -closure in X * * is a ball.
Theorem 56 [104] Let Ω be a locally compact Hausdorff space, and C 0 (Ω) be the Banach space of all continuous real valued functions which vanish at infinity on Ω endowed with the supremum norm. Then any nontrivial set K of constant width in C 0 (Ω) is a pseudoball.
It is shown in [47] that ( U 1 ) is strictly stronger than property (C) and strictly weaker than property (B).

Further related set families
Further related conditions are: The property that a set K satisfies (CD) does not imply that K is of constant width, and a complete set does not necessarily satisfy (CD) (cf. [105,Proposition 3.2]). In C(Ω) , where Ω is a compact Hausdorff space, each complete set satisfies (CD) (cf. [105,Proposition 3.3]). Example 5.2 in [118] shows that the property that K satisfies (CR+) does not imply that K is complete, and [118,Example 5.1] shows that the condition (CR) is strictly weaker than (CR+) since (CR) does not imply (D+). Example 4 in [28] shows that the condition (D+) does not imply Proposition 63 (cf. [118]) If a Banach space X is reflexive and smooth, then any K ∈ H satisfying (CD) is of constant width.
A convex body R is said to be reduced in X if Δ(C) < Δ(R) holds for any convex body C properly contained in R (cf. [75]). There are interesting open problems about complete and reduced bodies in Minkowski spaces (see the survey [77]), and even for the Euclidean norm surprisingly elementary questions are still unsolved (cf. [76]). It is interesting that the notion of reduced body, although it somehow dualizes that of completeness, yields a proper superset of the class of complete sets in the n-dimensional Euclidean space n . The reason is that in n the notions of constant width and completeness coincide, and that one can easily construct reduced bodies that are not of constant width. In Minkowski spaces, this coincidence of constant width and completeness is only assured for n = 2 (Meissner's theorem; see [93, p. 98]), and therefore in any Minkowski plane the class of reduced bodies clearly contains that of complete sets. Martini and Wu [94] showed that, when the dimension of the underlying space is at least 3, a complete set is not necessarily reduced. Richter [127] showed that every Minkowski space of dimension at least 3 contains reduced sets of arbitrarily large ratio of diameter and width. Since this ratio of complete sets in an n-dimensional Minkowski space is bounded from above by (n + 1)∕2 , every Minkowski space of dimension at least 3 contains reduced sets that are not complete. Richter showed further that every Minkowski space of dimension at least 2 contains reduced sets that are not complete.
Naturally one may ask whether a complete and reduced set has constant width. Brandenberg et al. [22] proved that if K is a complete and reduced set in a Minkowski space equipped with a smooth extreme point, then K has constant width. They also showed that a complete and reduced simplex in a Minkowski space has constant width. These results are stated and proved for a more general situation, see [22] for more details.
Domínguez Benavides and Papini [34] considered the notion of diametrical minimality: a set K ∈ H is said to be diametrically minimal if every closed, convex, proper subset L of K satisfies diam L < diam K . Moreno [103,Proposition 2.1] showed that diametrically minimal sets are segments. See [116] for relations between other related set families.

Interior and boundary
Concerning the circumradius and inradius of a complete set in a Minkowski space, Moreno and Schneider proved the following Theorem 64 [106] Let X be an n-dimensional Minkowski space, and K be a complete set in X . Then If equality holds in one of these inequalities, then K is a simplex.
Theorem 64 shows that any nontrivial complete set in a Minkowski space X has interior points. It also infers that when the underlying space is not finite-dimensional, there might exist nontrivial complete sets having empty interior. Actually, even nontrivial sets of constant width in infinite dimensional normed linear spaces may have empty interior. See the following example.
Example 5 (cf. [104]) In c 0 , the set is of constant width and has empty interior. This shows that [31, Proposition 3.2 and Corollary 3.7] are not correct.
In [83] it is shown that there exist reflexive LUR spaces having nontrivial complete subsets with empty interior. See [25] for more examples of such spaces. A necessary condition for a Banach space to contain a nontrivial complete set with empty interior is the following Let (X, ‖⋅‖) be an infinite dimensional and reflexive Banach space. If (X, ‖⋅‖) has the non-strict Opial property, then there exists an equivalent renorming (X, ‖⋅‖ 0 ) that has a nontrivial complete set with empty interior (cf. [62,63]). If X is separable (see [26, 27,  Therefore, nontrivial sets of constant width, complete sets, and sets of constant radius having empty interior only exist in spaces that lack of a normal structure. It is shown in [84] that, for any infinite dimensional normed linear space X and any ∈ (0, 1) , there exists a symmetric K ∈ H such that (1 − )B X ⊆ K ⊆ B X and that no boundary point of K is diametral.
A basic fact concerning the boundary of a complete set is the following Maluta [84] showed that there is an equivalent renorming of 2 and a set K of constant width such that boundary points of K not belonging to a diametral pair of K form a dense subset of bd K . She also showed that, in a reflexive space, the condition "every boundary point belongs to a diametral pair" does not characterize sets of constant width among bounded, closed, and convex sets with nonempty interior. It is shown in [85] that there exists a complete set K in c 0 such that equality in (7.2) holds.
Martín et al. [87] proved that, in a Minkowski space, the following holds:

Asymmetry of complete sets in Minkowski spaces
The starting point of this subsection is the following result saying that a complete set distinct from a ball is not symmetric.

Theorem 79 A centrally symmetric complete set in a normed linear space is a ball.
Proof Let K be a centrally symmetric complete set. Without loss of generality we may assume that diam K = 2 and K is symmetric with respect to o . To show that K is a ball, we only need to prove that K ⊆ B X since K and B X are complete sets having equal diameter. Otherwise, there exists x ∈ K⧵B X . It follows that −x ∈ K and diam K ≥ ‖−x − x‖ > 2 , a contradiction to the fact that diam K = 2 . ◻ In the book [140] of Toth two important topics from convexity are comprehensively treated: measures of symmetry and stability of geometric inequalities. Related to the first topic, we introduce the following notion. For a bounded closed convex set K in a normed linear space X , the Minkowski asymmetry s(K) of K is defined by A point c ∈ X satisfying is called a Minkowski center of K. Jin and Guo [56] showed that the most asymmetric sets of constant width in the Euclidean plane are Reuleaux triangles (cf. [60] for a "stable" version). They extended this result in [58] by showing that the most asymmetric sets of constant width in the n-dimensional Euclidean space are completions of n-dimensional regular simplices. In [59], Jin and Guo proved that completions of regular simplices are also the most asymmetric sets of constant width with respect to the mean Minkowski measure (cf. [140,Chapter 4]). The following result extends [57,Theorem 1].
Theorem 80 [20] If K is a complete set in a Minkowski space X , then .

(S)∕ diam S = n∕(n + 1).
Following [20], we say that a bounded closed convex set K is pseudo-complete if there exists c ∈ X such that A complete set is pseudo-complete, and the converse is not true.
Proposition 85 [20] If K is pseudo-complete in a Minkowski space, then 9 Structure of the space of diametrically complete sets A collection A of subsets of a normed linear space is called convex if it is called starshaped with respect to A if In [108] Moreno and Schneider studied properties of the family D of all diametrically complete sets in a given Minkowski space. They proved that in general this set is not convex, and also that it is not necessarily starshaped. Further on, they gave a characterization of norms in which this family is starshaped, and further results on polyhedral norms were also presented.
The Minkowski sum of two complete sets is not necessarily complete, see the following example.
The Minkowski sum (even the closure of the Minkowski sum) of a complete set and a ball is not necessarily complete (cf. [113] and [104,Example 5.3]). In 3 1 , the Minkowski sum of a complete set and a set of constant width is complete (cf. [104,Proposition 5.2]). Therefore, it is interesting to study the following conditions. Theorem 89 [110] Let > 0 . The family D of all complete sets in C(Ω) having diameter is starshaped with respect to the ball B C(Ω) (o, ∕2) . D is convex if and only if Ω is an extremally disconnected space.

Outlook
The concepts of complete and constant width sets are also interesting in further non-Euclidean geometries (see, e.g., pp. 240-243 in the monograph [88]). El-Kholy [37] and Dekster [33] studied non-Euclidean analogues of completeness in hyperbolic and spherical space. E.g., Dekster [33] proved that in hyperbolic space the concepts of complete and of constant width sets coincide (as in the Euclidean setting). And he showed that regarding spherical geometry, this characterization is only true in the convex case. Namely, in [33] he constructed examples of complete non-convex sets that are not of constant width, and he showed that complete sets of diameter larger than ∕2 can strongly differ from sets of the same constant width. Moreover, he verified that if a complete convex set has diameter larger than ∕2 , then it has to be smooth. Wegner [144] investigated certain variants of the completeness notion in spherical 2-space regarding their coincidence with the concept of constant width. With new proofs, also Lassak [73] studied the coincidence of completeness and constant width in spherical space. And the whole Section 2.11 of his interesting survey [74] on spherical geometry collects related basic results on complete sets in n-dimensional spherical space.