Uniform Convexity and Convergence of a Sequence of Sets in a Complete Geodesic Space

Abstract

In this paper, we first introduce two new notions of uniform convexity on a geodesic space, and we prove their properties. Moreover, we reintroduce a concept of the set-convergence in complete geodesic spaces, and we prove a relation between the metric projections and the convergence of a sequence of sets.

1. Introduction

There are a lot of works dealing with the relation between convergence of a sequence of sets and convergence of a sequence of projections corresponding to it. In particular, the following theorem on a reflexive and strictly convex real Banach space is one of the important results.

Theorem 1 (Tsukada [1]). Let be a strictly convex and reflexive real Banach space satisfying the Kadec-Klee property. Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset such that converges to in the sense of Mosco. Then converges strongly to for any , where is the metric projection of onto a nonempty closed convex subset of .

Since a uniformly convex real Banach space is strictly convex, reflexive, and satisfying the Kadec-Klee property, this theorem is true in uniformly convex real Banach spaces. Moreover, since a real Hilbert space is a uniformly convex real Banach space, it is also true in real Hilbert spaces.

On the other hand, we know that a Hadamard space is another generalization of Hilbert spaces. It is defined as a complete metric space having a particular convexity structure and it also has various useful properties that Hilbert spaces have. Kimura [2] introduced -Mosco convergence in complete geodesic spaces using a notion of asymptotic centre instead of weak convergence.

Theorem 2 (Kimura [2]). Let be a complete CAT(0) space. Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset such that converges to in the sense of -Mosco. Then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

Moreover, Kimura and Satô [3] introduced -Mosco convergence in complete CAT() spaces and obtain the following result:

Theorem 3 (Kimura and Satô [3]). Let be a complete admissible CAT() space for . Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset such that converges to in the sense of -Mosco. Then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

In this work, we introduce a new concept of the set-convergence and we obtain a similar result as above under the assumptions that both uniformly convex real Banach spaces and complete CAT(0) spaces have.

2. Preliminaries

A function is said to be a gauge if is strictly increasing, continuous, and . We know that if is a real sequence of such that for some gauge function , then .

Let be a real Banach space. Then the following propositions are equivalent: (i) is uniformly convex(ii)For , , and , it holds that whenever(iii)For any , there exists a convex gauge function such that

for any , where and for

For more details about the properties of uniformly convex real Banach spaces, see [4].

Let be a metric space and let . A geodesic path from to is a mapping such that , and for any . Let . If for any such that , a geodesic path from to exists, then we say that is -geodesic. Moreover, if such a geodesic path is unique for each pair of points, then is said to be -uniquely geodesic. In a -uniquely geodesic space, the image of a geodesic path from to is called a geodesic segment joining and and it is denoted by . For and , there exists a unique point such that and . We denote it by . A geodesic triangle with vertices is the union of geodesic segments , and . We denote it by .

To define a CAT() space, we use the following notation called a model space. For , the two-dimensional model space is the two-dimensional Euclidean space with the metric induced from the Euclidean norm. For , is the two-dimensional sphere whose metric is a length of a minimal great arc joining each two points. For , is the two-dimensional hyperbolic space with the metric defined by a usual hyperbolic distance.

The diameter of is denoted by , defined by

We know that is a -uniquely geodesic space for each .

Let . For in a geodesic space satisfying that , there exist points such that , , . We call the triangle having vertices , and in a comparison triangle of . Notice that it is unique up to an isometry of . For a specific choice of comparison triangles, we denote it by . A point is called a comparison point for if .

Let and a -geodesic space. If for any with , for any , and for their comparison points , the CAT() inequality holds, then we call a CAT() space. It is well known that any CAT() space is also a CAT() space whenever . Therefore, a CAT() space is a CAT(0) space for any .

Let be a CAT() space for . If for any , then we say that is admissible. A CAT() space is always admissible when .

A subset of a -uniquely geodesic space is said to be convex if for every and . For a subset of -uniquely geodesic space , a closed convex hull of is defined as the intersection of all closed convex sets including S, and we denote it by clco S.

Let be a complete admissible CAT() space for and a nonempty closed convex subset of . Then for , there exists a unique point such that

We call such a mapping defined by , the metric projection of onto .

Let be a CAT(0) space. From the CAT(0) inequality, it is easy to see that for every and .

The following lemma shows that a CAT(0) space has a similar property to the uniform convexity of Banach spaces.

Lemma 4. Let be a CAT(0) space. For , a point and two sequences , it holds that whenever

Proof. For , and , if then which implies Letting , we have This is the desired result.

Let be a metric space. For a point and a nonempty subset , the distance between them is defined by .

Let be a metric space. A function is said to be lower semicontinuous if for and , whenever . Moreover, a function is said to be proper if there exists a point such that and the domain of defined by

Let be a uniquely geodesic space. A function is said to be convex if for and , satisfies

3. Uniform Convexity of a Complete Geodesic Space

In the following, we always suppose that for any in geodesic space , a geodesic joining to is unique.

A geodesic space is said to satisfy the condition (D) if: for and .

We introduce two new concepts of uniform convexity on a geodesic space.

Let be a geodesic space. is said to be sequentially uniformly convex if satisfies the condition (D) and, for , a point and two sequences , it holds that whenever

Let be a geodesic space. is said to be uniformly convex if for any there exists a convex gauge function such that for any , where with and .

Uniformly convex real Banach spaces and CAT(0) spaces are uniformly convex in this sense.

Theorem 5. Let be a uniformly convex geodesic space. Then, is sequentially uniformly convex.

Proof. Let and let . Then there exists a convex gauge function , satisfying that for any and hence That is, satisfies the condition (D).
Moreover, for , and , if then there exists such that for any . From uniform convexity of , there exists a convex gauge function , satisfying that It follows that Therefore, we obtain and hence , that is, is sequentially uniformly convex.

Theorem 6. Let be a sequentially uniformly convex geodesic space. For and with , if, then .

Proof. For and with , we suppose that . If , then, since we have . From the sequential uniform convexity of , we have . This is a contradiction. Therefore, we have . This is the desired result.

Theorem 7. Let be a sequentially uniformly convex complete geodesic space and let be a nonempty closed convex subset of . Then, for , there exists a unique point such that .

Proof. For , let . Then, for , we can take a sequence such that Then, we have . Suppose that is not a Cauchy sequence. That is, there exists such that for any , there exist such that . In this way, we take two subsequences . Then, and we have Hence, from the sequential uniform convexity of , we have This is a contradiction and thus is a Cauchy sequence. Since is complete and is closed, there exists such that . Therefore, we have Next, we show the uniqueness of . Suppose that satisfy and . Then, from Theorem 6, we have This is a contradiction. Therefore, for , there exists a unique point such that .

Let be a sequentially uniformly convex complete uniquely geodesic space and let be a nonempty closed convex subset of . Then for , there exists a unique point such that

We call such a mapping defined by , the metric projection of onto .

Theorem 8. Let be a sequentially uniformly convex complete geodesic space and a sequence of nonempty bounded closed convex subsets which is decreasing with respect to inclusion, that is, for any . Then, is nonempty.

Proof. Since is nonempty bounded closed convex subset for , for , we can take a sequence by , where is the metric projection of onto a nonempty closed convex subset of . Then is a bounded increasing real sequence and hence has a limit . That is, we have First, we show that converges to some point . If , then, since , we have as . Hence, we may suppose that . Suppose that is not a Cauchy sequence. That is, there exists such that for any , there exist such that . Without loss of generality, we can suppose that . In this way, we take two subsequences . Then, Since , we have and thus From the sequential uniform convexity of , we have . This is a contradiction. Therefore, is a Cauchy sequence and thus there exists such that .
We show that . For , for and thus . Therefore, and it completes the proof.

4. -Convergence

Let be a metric space and a bounded sequence. An asymptotic centre of is defined by

Lemma 9. Let be a sequentially uniformly convex complete geodesic space and let be a bounded sequence. Then, there exists a point such that That is, is nonempty. Moreover, is bounded, closed and convex.

Proof. Let and define by for any . Then, for any , is nonempty and bounded. If and , then Letting , we have and hence . Moreover, for and , we have It implies . Therefore, is closed and convex for any . Moreover, is decreasing with respect to inclusion. Hence, we have and thus Also, we know that is closed and convex.

If we suppose uniform convexity for , we can prove that an asymptotic centre is a singleton.

Theorem 10. Let be a uniformly convex complete geodesic space and let be a bounded sequence of . Then, is a singleton.

Proof. Let . If with , then, since is bounded, there exists such that and for all . Then, there exists a convex gauge function , and we have and hence Since , we have and thus This is a contradiction. Therefore, is a singleton.

Let be a bounded sequence and . We say -converges to a-limit if is the unique asymptotic centre of any subsequences of , and we denote it by .

Let be a geodesic space. is said to satisfy the condition (C), if any nonempty closed convex subset is -closed, that is, if and , then .

CAT(0) spaces satisfy the condition (C). Let be a uniformly convex real Banach space. Then, the following propositions are equivalent: (i)For any bounded sequence , -converges to if and only if converges weakly to (ii) satisfies the condition (C)(iii) satisfies Opial’s condition

See [5] for details.

The following two theorems can be proved by the same method as the corresponding results in [6–8].

Theorem 11 (Bačák [6], Kirk and Panyanak [8]). Let be a uniformly convex complete geodesic space. Then, for any bounded sequence of has a -convergent subsequence.

Corollary 12. Let be a uniformly convex complete geodesic space satisfying the condition (C) and let be a nonempty bounded closed convex subset of . Then, for any sequence in has a -convergent subsequence and its -limit belongs to .

Theorem 13 (Bačák [6], He, Fang, Lopez and Li [7]). Let be a uniformly convex complete geodesic space satisfying the condition (C). Let be a proper lower semicontinuous function and a sequence such that . Then,

Corollary 14 (-lower semicontinuity of the distance function). Let be a uniformly convex complete geodesic space satisfying the condition (C) and let be a point. Let be a sequence such that . Then,

Lemma 15. Let be a uniformly convex complete geodesic space and let be a sequence such that -converges to . Then,

Proof. Since is bounded, so is . For any subsequence of , if , then, we have Therefore, since , we have . Hence, is the unique asymptotic centre of any subsequence of and it completes the proof.

Theorem 16 (-Kadec-Klee property). Let be a uniformly convex complete geodesic space satisfying the condition (C) and a sequence such that and for some . Then, .

Proof. Let be a sequence such that for any . Since , From Lemma 15, we have . Then, since is -lower semicontinuous and , we have and thus . Therefore, we obtain If , then . If , from sequential uniform convexity of , we obtain . Since for any , we have .

5. Convergence of a Sequence of Sets

Let be a sequence of nonempty closed convex subsets of a uniformly convex complete geodesic space . -Mosco convergence is defined by using a notion of asymptotic centre by Kimura [2]. First, we define subsets and of as follows: if and only if there exists such that and for all ; if and only if there exist a bounded sequence and a subsequence of such that and for all . If a subset of satisfies that , we say that converges to in the sense of -Mosco.

Here, we introduce a new concept of the set-convergence. We define subsets of as follows: if and only if there exists a bounded sequence and a subsequence of such that and for all .

Since a convergent sequence is a -convergent sequence, the inclusion is always true. If a subset of satisfies that , we say that converges to in the sense of -Mosco. Furthermore, the following inclusion holds:

Therefore, if converges to in the sense of -Mosco, then converges to in the sense of -Mosco.

Lemma 17. Let be a uniformly convex complete geodesic space satisfying the condition (C) and a sequence of nonempty closed convex sets which is decreasing with respect to inclusion, that is, for any . if is nonempty, then converges to in the sense of -Mosco.

Proof. We show that . If , then for any . Let be a sequence of such that for any . Then, and for any . Therefore, we obtain and hence .
Next, we show that . If , then there exists a sequence and such that for any and . For any , if , then . Therefore, from the condition (C) of , we have . Since is arbitrarily, we obtain . Hence, we have .
Since is always true, we have That is, converges to in the sense of -Mosco.

Theorem 18. Let be a uniformly convex complete geodesic space satisfying the condition (C). Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . If converges to in the sense of -Mosco, then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

Proof. Fix arbitrarily. Since , there exists a sequence such that and for every . Since for any and is bounded, is also bounded. Moreover, letting , we have Here, we take a subsequence of arbitrarily. Since is bounded, there exists a subsequence of and such that Let for any . Then, since for any there exists such that for any , it holds that Let . Then we have for any . Since is closed and convex, from the condition (C) of , we obtain . Therefore, we have Letting , we obtain Since , we have and thus Hence, we obtain . Since is -lower semicontinuous and , we have and hence Therefore, . Since and , from the -Kadec-Klee property of , we obtain . Since for any subsequence of , there exists a subsequence of such that , we have .

Let be a uniformly convex complete geodesic space. If has the condition (C), we can show that convergence of a sequence of metric projections from -Mosco convergence. If a sequence of sets which is decreasing with respect to inclusion, then we can show it on sequentially uniformly convex complete geodesic spaces without the condition (C).

Theorem 19. Let be a sequentially uniformly convex complete geodesic space and a sequence of nonempty closed convex sets which is decreasing with respect to inclusion, that is, for any . Suppose that is nonempty. Then, converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

Proof. Fix arbitrarily. Since , if , then for any , we have and hence is bounded. Furthermore, since for any , we have and thus is increasing. Therefore, has a limit First, we show that converges to some point . Since if , we may assume that . Suppose that is not a Cauchy sequence. That is, there exist such that for any , there exist such that . Without loss of generality, we can suppose that . In this way, we take two sequences . Then, and satisfies Since , we obtain Letting , we obtain From sequential uniform convexity of , we have This is contradictory to and hence is a Cauchy sequence. Therefore, since is complete, there exists a point such that .
Next, we show that . For any , since if and is closed, we have and thus .
Finally, we show that . For any , form the property of the metric projection, we have Letting , we obtain and hence .
Consequently, we have for any .

6. Convergence of a Sequence of Sets in CAT() Spaces

In this section, we will consider CAT() spaces for .

Let be a metric space and let be a bounded sequence of . The asymptotic radius of defined by

Let be a sequence of nonempty closed convex subsets of a complete admissible CAT() space for . -Mosco convergence is defined by Kimura and Satô [3]. First, we define subsets and of as follows: is the same as in the case of uniformly convex complete geodesic spaces, that is, if and only if there exists such that and for all ; if and only if there exist a sequence and a subsequence of such that , and for all . If a subset of satisfies that , it is said that converges to in the sense of -Mosco. The notion of -Mosco convergence coincides with -Mosco convergence in complete CAT() spaces for .

We define a subset of as follows: if and only if there exist a sequence and a subsequence of such that , and for all . If , then .

Since a convergent sequence is a -convergent sequence, the inclusion is always true. If a subset of satisfies that , it is said that converges to in the sense of -Mosco. Furthermore, the following inclusion holds:

Therefore, if converges to in the sense of -Mosco, then converges to in the sense of -Mosco.

-Mosco convergence coincides with -Mosco convergence in complete CAT() spaces for .

Since complete CAT() space is uniformly convex and satisfies the condition (C) for any , we have the following theorem:

Theorem 20. Let be a complete CAT() space for . Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . If is converges to in the sense of -Mosco, then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

We also have the following result [2].

Theorem 21 (Kimura [2]). Let be a complete CAT() space for and is the metric projection of onto a nonempty closed convex subset of . Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . If converges to for any , then converges to in the sense of -Mosco.

Let be a complete CAT() space and is the metric projection of onto a nonempty closed convex subset of . Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . Then, from Theorem 19 and Theorem 21, the following propositions are equivalent: (i) converges to in the sense of -Mosco(ii) converges to in the sense of -Mosco(iii) converges to for any

That is, the notion of -Mosco convergence coincides with -Mosco convergence in complete CAT() spaces for .

Since most of the results for CAT() spaces for are easily deduced from that for CAT(1) spaces, in the following, we focus on CAT(1) spaces.

We first see the following known results for CAT(1) spaces.

Theorem 22 (Espínola and Fernádez-León [9]). Let be a complete CAT(1) space. If a sequence satisfies , then there exists a -convergent subsequence of .

Theorem 23 (Espínola and Fernádez-León [9]). Let be a complete admissible CAT(1) space and let be a sequence satisfies . Then,

Theorem 24 (He, Fang, Lopez, and Li [7]). Let be a complete CAT(1) space and let be a sequence such that . Then for any with , the following inequality holds:

Theorem 25 (Kimura and Satô [3]). Let be a complete admissible CAT(1) space and let be a sequence. Suppose that and . Then .

Using these results, we obtain the relation between -Mosco convergence of sets and corresponding sequence of metric projections.

Theorem 26. Let be a complete admissible CAT(1) space. Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . If converges to in the sense of -Mosco, then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

Proof. Fix arbitrarily. Since , there exists a sequence such that . Since for any . Moreover, letting , we obtain Here, we take a subsequence of arbitrarily. From Theorem 22, there exists a subsequence of and such that Let for any . Then, since for any there exists such that for any , it holds that Let . Then we have for any . Since is closed and convex, from Theorem 23, we have . Therefore, we have Letting , we obtain Since , we have and thus Hence, we obtain . From Theorem 24, we have and hence Therefore, . Since and , from Theorem 25, we have . Since for any subsequence of , there exists a subsequence of such that , we have .

We also have the following result [3].

Theorem 27 (Kimura and Satô [3]). Let be a complete admissible CAT(1) space and is the metric projection of onto a nonempty closed convex subset of . Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . If converges to for any , then converges to in the sense of -Mosco.

Let be a complete admissible CAT(1) space and is the metric projection of onto a nonempty closed convex subset of . Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . Then, from Theorem 26 and Theorem 27, the following propositions are equivalent: (i) converges to in the sense of -Mosco(ii) converges to in the sense of -Mosco(iii) converges to for any

That is, -Mosco convergence coincides with -Mosco convergence in complete admissible CAT(1) spaces.

For the case where , by using standard modification, we can obtain the same result. That is, -Mosco convergence coincides with -Mosco convergence in complete admissible CAT() spaces for .

Consequently, -Mosco convergence coincides with -Mosco convergence for every and we can replace the definition. That is, if a subset of a complete admissible CAT() space for satisfies that then we can say that converges to in the sense of -Mosco.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by JSPS KAKENHI Grant Number JP21K03316.