On Some Properties of Hyperconvex Spaces
© Marcin Borkowski et al. 2010
Received: 13 September 2009
Accepted: 13 January 2010
Published: 24 February 2010
We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete -trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for -trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.
It is hard to believe that although hyperconvex metric spaces have been investigated for more that fifty years, some basic questions in their theory still remain open (let us recall that hyperconvex metric spaces were introduced in  (see also ), but from formal point of view it has to be emphasized that the notion of hyperconvexity was investigated earlier by Aronszajn in his Ph.D. thesis  which was never published). The main purpose of this paper is to answer some of these questions.
Let us begin with the notion of hyperconvex hull which was introduced by Isbell in  (see Definition 2.7). This notion is more difficult to investigate than the classical notion of convex hull, since the former one is not uniquely determined (see Proposition 2.8). In Section 3 we are going to prove that in hyperconvex metric spaces with the unique metric segments property, hyperconvex hulls are uniquely determined. Let us recall that such hyperconvex spaces were characterized by Kirk (see ) as complete -trees (see Theorem 2.15). This led to a surprising application of the theory of hyperconvex spaces to graph theory (see ).
Another interesting question is about the relation between the notion of convexity and hyperconvexity (cf. Remark 4.1). In particular, it is inspired by the following Sine's remark [7, page 863], stated without a proof: "The term hyperconvex does have some unfortunate aspects. First, a hyperconvex subset of even (with the norm) need not be convex. Also convex sets can fail to be hyperconvex (but for this one must go to at least )." It turns out that all hyperconvex subsets of a given normed space are convex if and only if the space in question is strictly convex; this fact is proved in Section 4.
In Section 5 we turn our attention to the classical Krein-Milman theorem (see ). We prove that a bounded complete -tree is a convex hull of its extremal points (note that a similar result, but with the assumption of compactness, is proved in ). Hence, in particular, such a property holds for bounded hyperconvex metric spaces with unique metric segments.
Let us denote by and the Kuratowski and Hausdorff measures of noncompactness, respectively, (see [10, 11] for the definition and basic properties). It was noticed by Espínola (see ) that if a metric space is hyperconvex, then for all its bounded subsets . The question is about the inverse implication. More precisely, assume that for every bounded subset of a given metric space . Does this equality imply that is hyperconvex? (Obviously, we mean nontrivial cases, i.e., we exclude spaces in which every bounded set is relatively compact.) In Sections 6 and 7 we introduce a few metric spaces which are not hyperconvex, but for all their bounded subsets. Hence the answer to the above question is negative. Let us emphasize that the metrics considered in Sections 6 and 7 are extensions and generalizations of commonly known radial metric and river metric, which were proved in  to be hyperconvex.
Let us notice that in general it is not easy to provide explicit formulae which would allow to evaluate the measures of noncompactness in particular spaces. We are going to state such formulae for the metric spaces considered in Sections 6 and 7.
Let us emphasize that another motivation to consider those metrics comes from the real world. Let us consider an example of the transmission of phone signals, when one person (say, ) calls another (say, ), assuming there are two base transceiver stations (say, and ). We may have two cases. If and are in the range of one of the BTS's, say , then the signal is first transmitted from to and then from to —even if and are "close" to each other. If and are located in the ranges of and respectively, then the signal is transmitted from to , then from to and finally from to . Hence we have the metric considered in Definition 7.4.
In Section 8 we provide a general scheme to construct metrics similar to these of Sections 6 and 7. This scheme is a generalization of a construction from .
For completeness, in Section 2 we collect some basic definitions and facts used in the sequel.
Let us begin with some classical definitions and facts.
Definition 2.2 (see [1, page 410, Definition 1]).
Hyperconvex spaces possess—among others—the following properties.
Proposition 2.3 (see [1, page 417, Theorem 1']).
A hyperconvex space is complete.
Proposition 2.4 (see [1, page 423, Theorem 9]).
A nonexpansive retract (i.e., a retract by a nonexpansive retraction) of a hyperconvex space is hyperconvex.
Proposition (see [1, page 422, Corollary 4]).
Each hyperconvex metric space is an absolute nonexpansive retract, that is, it is a nonexpansive retract of any metric space it is isometrically embedded in. In particular, hyperconvex spaces are absolute retracts.
The following theorem gives a characterization of hyperconvex real Banach spaces.
A real Banach space is hyperconvex if and only if it is isometrically isomorphic to some space of all real continuous functions on a Hausdorff, compact and extremally disconnected topological space with the norm.
Now let us state the definition of a hyperconvex hull. We will not need the general version of this notion, investigated by Isbell in ; instead, the notion of a hyperconvex hull of a subset of a hyperconvex space will suffice for our considerations.
Definition 2.7 (see, e.g., [17, page 408]).
A hyperconvex hull always exists, but needs not to be unique. It is, however, unique up to an isometry. To be more precise, the following holds.
Proposition 2.8 (cf. [17, page 408, Proposition 5.6]).
Each nonempty subset of a hyperconvex metric space possesses a hyperconvex hull. If and are hyperconvex spaces, , are isometric and is an isometry, then for any hyperconvex hulls , of and respectively, the isometry extends to an isometry .
In what follows, we will also need the definitions of total and strict convexity.
A metric space is called totally convex if for any two points and for all such that there exists a point satisfying the equalities and . If this point is unique for all possible combinations of , we call the space convex and denote this point by .
Remark 2.10 (see [1, page 410]).
A hyperconvex space is totally convex.
Remark 2.11 (see, e.g., [18, page 7]).
For normed spaces, the above definition of strict convexity (Definition 2.9) coincides with the usual one.
Proposition 2.12 (see, e.g., [18, page 7]).
In a strictly convex metric space, intersection of any family of totally convex subsets is itself totally convex.
The above proposition lets us define the notion of a convex hull in any strictly convex metric space in a natural way.
Definition 2.14 (see, e.g., [5, page 68, Definition 1.2]).
(Let us note that (3) follows from (1); it is, however, useful to have it among the basic properties of -trees.) We will also use the notation for an open metric segment joining and and for a left-open one.
Theorem 2.15 (see [5, Theorem 3.2]).
In what follows, we will also use the classical notions of Chebyshev subset of a metric space, a metric projection onto such a set (which we will denote by ), Kuratowski and Hausdorff measures of noncompactness (which we will denote by and , resp.), and the radial and river metrics (which we will denote by and , resp.). The reader may find the relevant definitions, for instance, in the papers [11, 19, 20].
For , it is enough to use Proposition 2.3 and Remark 2.10. On the other hand, if a subset of an -tree is closed and totally convex, it is a complete sub- -tree of . Indeed, it is enough to show that for each , the metric segment . But in view of the strict convexity of , we have . Now, in view of Theorem 2.15, is hyperconvex.
A natural question to ask is: in which hyperconvex metric spaces the hyperconvex hulls are unique? The following theorem answers this question.
Necessity follows easily from Proposition 3.1 and Theorem 2.15. Sufficiency. Let be a subset of an -tree . Notice that . Using Propositions 3.2, 2.12 and 3.3, we arrive at the conclusion that is the hyperconvex hull of in .
4. Normed Spaces
In the first part of this section we will give an answer to the following question: In which spaces closed and convex subsets are hyperconvex?
Note that the question whether all closed and convex subsets of some normed space are hyperconvex makes sense only in spaces which are themselves hyperconvex, so we will now restrict our attention to such spaces.
Theorem 4.2 (see [21, page 474, Theorem 1]).
Notice that "any hyperconvex norm on " means essentially (i.e., up to an isometric isomorphism) the maximum norm; this follows from Theorem 2.6 and can also be proved using a geometric argument (see [19, Theorem 4.1]).
Since is not isometrically isomorphic to , its dimension must be at least 2. Further, since the only (up to an isometric isomorphism) two-dimensional hyperconvex space is , we may assume . By Theorem 2.6 we may assume that is the space for some Hausdorff, compact and extremally disconnected topological space . Since , the space has at least three points, so includes a copy of . This means that it is enough to prove the theorem in case of with the "maximum" norm.
For simplicity, we will construct an affine non-hyperconvex subspace of ; by an appropriate translation one can obtain a linear one. Let . Consider the following three balls in : , , . Since the corresponding balls in intersect only at the space is not hyperconvex.
Corollary 4.3 and Theorem 4.5 yield the following characterization.
We will now turn our attention to the problem of describing the spaces in which hyperconvexity implies convexity. We will start with an observation suggested to us by Grzybowski .
Let be at least two-dimensional. Therefore there exist three noncollinear points . Put and let , . It is clear that and similarly for other distances. But is strictly convex, so we have and , so . It must be therefore , which finishes the proof.
From Proposition 4.7 we know that hyperconvex subsets of are one dimensional; but from Proposition 2.5 we infer that hyperconvex sets are connected, which for one-dimensional sets is equivalent to their convexity.
To prove the inverse implication, we will need a simple lemma.
Lemma 4.9 (see [23, page 44, Lemma 15.1]).
Now we are ready to prove the following theorem.
Assume that is not strictly convex; we will construct a nonconvex, hyperconvex subset of . There exist points and positive numbers such that , and the equalities and hold. From Lemma 4.9, both sets and , where means an affine segment with endpoints , , are metric segments joining and (and hence hyperconvex sets). They cannot be, however, both convex, so at least one of them is the desired counterexample.
Again, combining Corollary 4.8 and Theorem 4.10, we obtain the following characterization of strictly convex normed spaces.
A normed space is strictly convex if and only if each its hyperconvex subset is convex.
5. Krein-Milman Type Theorem
It is enough to show that each point of lies on a metric segment joining some two extremal points of . Let . We may assume that is not extremal; let . The family of all metric segments having as one of its endpoints satisfies the assumptions of the Kuratowski-Zorn lemma. Let and be maximal metric segments containing the respective given metric segments. We will first show that and are extremal points.
Since closed and convex subsets of an -tree are hyperconvex (Proposition 3.3), Corollary 4.6 might give the impression that -trees are somehow similar to 1- or 2-dimensional vector spaces and that completeness and boundedness of an -tree imply its compactness. As the following example shows, this analogy is misleading.
6. Hyperconvexity and Measures of Noncompactness
Let us begin this section with the following definition.
It is easy to prove the following lemma.
Now we are going to examine the measures of noncompactness in the space . For this purpose we are going to use a similar approach as in the case of the measures of noncompactness in with the radial metric (cf. [20, Theorem 4]). First let us introduce the following definition.
Using above conditions we can prove the following theorem
The above theorem shows that even in the nontrivial cases (i.e., in cases, when bounded sets are not necessarily relatively compact), the above equality does not have to imply that the space in question is hyperconvex.
Definition 6.1 can be slightly modified. Namely, let us introduce the following definition.
It can be easily checked that is a complete metric space. Its topology is also stronger than the topology of with the radial metric. On the other hand this topology is obviously equivalent to the topology induced by the metric .
The proof of Theorem 6.13 is similar to the proof of Theorem 6.6 and therefore we omit it.
The metric we are going to consider to the end of this section is, roughly speaking, like between the radial metric and the river metric. We will call it a modified river metric.
The following fact can be easily checked.
7. Generalized Modified Radial and River Metrics
The metric spaces as well as are special cases of a general construction provided in . More precisely, let be a normed space and its Chebyshev subset.
The above defined function is a metric (see [19, Lemma 3.1]). Now, the following question can be risen. Is it possible to consider two disjoint Chebyshev sets, instead of one Chebyshev set , in such a way to get a variant of the metric defined above? The following two examples show that in the case of classical hyperconvex metrics: the radial metric as well as the river metric, this problem seems not to be easy.
for all , where denotes the river metric. Then this is not a metric. Indeed, it does not satisfy the triangle inequality in the following case. Let and let us take three points Then, by the definition and but , which shows
However, it appears that all the metrics introduced in Section 6 (Definitions 6.1, 6.9 and 6.14) are appropriate to define new metrics using the idea described at the beginning of this section.
Let us begin with the following definition
It is easy to check that is a metric. Now to verify that it is complete, let us consider a Cauchy sequence in the space . Then there exists such that for all , the points belong to the same closed half-plane or . Hence, by Lemma 6.2, is convergent, which completes the proof.
Now, using the metric from Definition 6.14, let us introduce the following metric.
One can prove the following lemma.
The proof of this Lemma is similar to the proof of Lemma 7.5 and therefore we omit it.
The metric is a variant of the metric defined in Definition 6.14. The topologies induced by these metrics are not comparable. The space is not hyperconvex, either. Finally, to find the Kuratowski and the Hausdorff measures of noncompactness of bounded sets in with the metric , it is enough to use the same approach as in Remark 7.8.
In Definitions 7.4 and 7.10, we considered two Chebyshev sets. Now one can think of the following question. Is it possible to increase the number of suitably chosen Chebyshev sets? The answer is "yes." Let us introduce the following definition.
So, if we choose then which contradicts that is a Cauchy sequence. Hence almost all the terms of any Cauchy sequence must be in the same closed quadrant. Thus by Lemma 6.2, is convergent, which shows that the space is complete.
Obviously, the following lemma holds.
8. Linking Construction
In this section we will give a slight generalization of the so-called linking construction described by Aksoy and Maurizi in  and show how this generalization includes the metrics of Section 7. Notice that a similar concept appears in , where it is used to study existence of certain mappings between Banach spaces.
Definition 8.1 (cf. [14, page 221, Theorem 2.1]).
Theorem 8.2 (cf. [14, page 221, Theorem 2.1]).
The paper  contains the above theorem only for hyperconvex spaces. It is obvious that is a metric also in the general case.
The authors of the paper  applied their version of Theorem 8.2 to obtain the hyperconvexity of the metric of Definition 7.1 (see [14, Theorem 2.2]). Let us notice that an identical result was given in an earlier work .
At the beginning of Section 7 we posed a question whether it is possible to construct a metric analogous to that from Definition 7.1, but with more than one Chebyshev subset. In all our examples, however, these subsets were singletons. Let us now show an example of two similar metrics constructed using two disjoint Chebyshev subsets consisting of more than one point.
For each , let . Let the identity map and be defined by for . The metrics on and 's are inherited from . Applying Definition 8.1 we obtain a certain metric on . Let us notice that it is not complete; taking and for , and as before we obtain another metric, this time complete. Let us finish by observing that since , and hence , is disconnected, in both cases cannot be hyperconvex.
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