Mutually Nearest Points for Two Sets in Metric Spaces

In this paper, we give some new conditions for the existence and uniqueness of mutually nearest points of two sets, i.e., two points which achieve the minimum distance between two sets in a metric or Banach spaces. These conditions are seen in case of compact sets, weak Compact sets, closed and convex sets, weakly sequentially compact sets and boundedly compact sets and their combinations. The study is confined to metric spaces and normed or Banach spaces. Some geometric properties of a Banach spaces like; strictly convexity, uniformly convexity, P-Property and weak P-Property are introduced. Also, we introduce the concept of generalized weak P-Property and give some interesting results. The present work may be briefly outlined as follows: It is the mathematical study that is motivated by the desire to seek answers to the following basic questions, among others. Which subsets are mutually proximinal? How does one recognize when given elements x ∈ A and y ∈ B are the nearest points of A and B? which is called a natural extension of the best approximation problem. Can one describe some useful algorithms for actually computing nearest points between two given sets? And how to find closely related sets to the proximity maps.


Introduction
The problem of existence and uniqueness of mutually nearest points has been studied by many investigators. In normed linear spaces the conditions under which two sets are distance sets have been studied by Bor-Luh-Lin [15], Dionisio [16], Tukey [17], Klee [18], Pai [19] and others. An application of distance sets to linear inequalities has been cited by Cheney and Goldstein [20]. The same problem in metric spaces has been studied by Nicolescu [21] and is also discussed by Singer [22]. The nearest point and its existence and uniqueness property has been introduced in [1,2] with the use of weak P-Property and generalized weak P-property. A natural extension of the nearest point problem is to find mutually nearest points relative to two sets A, B in a metric space or Best proximity pair evolves as a generalization of the concept of best approximation. Several authors have studied such mutually nearest points. For example; mutually nearest points relative to two closed convex sets in Hilbert spaces are found in many literatures. Mutually nearest points in reflexive Banach spaces is shown by Chong Li [10]. In this paper, the authors have also studied different conditions under which two sets may be distance sets.

Some Definitions and Notations
Definition 2.1. [4] Let A and B be any two nonempty subsets in a metric space X. Then the distance between two sets A and B is defined by But our Problem is to find the mutual nearest points, i.e., given subsets A, B in the metric space (X, d), we find We call (a, b) a best proximity pair of A and B. If there exists the pair (a, b) that satisfies the Equation (1), then we say that (a, b) is the solution of Equation (1). Best approximating pairs in a metric space may not exist in general. If B reduces to a singleton set {x} then the Problem (1) reduces to the problem of best approximation when A∩ B ≠ ϕ and it reduces to the well known feasibility problem for two sets and its solution set is The Formulation (2) captures a wide range of problems in applied mathematics and engineering. It is also used in the solution of linear inequalities, determining separating hyper planes, existence of fixed point and the method of alternating projections [14] etc. Definition 2.2. Let A, B be nonempty bounded sets in a Banach space X. We define Clearly, then the pair (x . , y . ) is called a best proximity pair (mutually nearest points) for A and B.
is nonexpansive (and so continuous on A).
Proof. Let u, v ∈B and a∈ A. Then By symmetry, we have Combining (3) and (4) we get, That is, Hence the distance function DA is non-expansive (and so continuous on A. Indeed, it is uniformly continuous.).
Recall [5] that any sequence (a n , b n ) with a n ∈A and b n ∈ B for all n such that lim n→∞ || a n -b n || = d(A, B) is called a minimizing sequence for (a, b) ∈ A × B.
Lemma 2.5 Suppose that A, B are closed subsets of a Banach space X. If some minimizing sequence{(x n , y n )} ⊆A× B has a weak cluster point (x, y) ∈A× B, then Suppose on the contrary that there exists (x . , y . ) ∈ (A, B) such that Moreover, there is δ ∈(., d(x, A)) such that B X (x . , δ )⊆A and B X (y . , δ )⊆B. Put x′= x . + δ (y . -x . )/2||y . -x . ||∈B X (x . , δ ) and Then x′ ∈A and y′ ∈B. Hence This contradicts (5) and the proof is complete.

Existence of Mutually Nearest Points
Theorem 3.1. [9] Let A be a compact set in ℝ n and let B be a closed subset of ℝ n . Define d(A, B) = inf x∈ A, y∈B ||x -y|| Show that there exist elements x . ∈ A and y . ∈ B such that.
, x∈A is continuous and attains its minimum for some x . ∈ A, so there exists y . ∈ B such that.
and the reverse inequality, is obvious.
But in this section we will extend this result.
Proof. The distance function d: X×X → ℝ is continuous (see Theorem 2.3), and the set A × B is compact in X × X by Corollary7.11. Therefore the Extreme Value Theorem applies and indicates that d takes on a minimum value on A×B. That is, there exist a * ∈A and b * ∈B such that.
It follows that.
Proof. By definition of an infimum, we can find a sequence of pairs ((u n , v n )) in A× B: d(u n , v n ) >a +1/ n for each n ≥1.
Since we are in a metric space, we can use sequential compactness to take subsequences of each sequence and find respective limits x∈ A and y ∈ B. Using the triangle inequality, we have for any δ >0 Theorem 3.4. Let A and B be two disjoint compact sets in a metric space (X, d). Then there exists x ∈A and y ∈ B such 2. The conclusion does not follow even assuming the completeness of the under-lying metric space. Consider the space ℓ p of sequences (say, p = 2), and let A = {a n }, where a n = (0, 0,⋯, 1 + 1/ n, 0, 0,⋯) Boundedness of A is clear, it is also closed since it has no limit points. However, for b = 0, the distance d(b, a n ) gets arbitrarily close to, but never, 1.
3. Let X be a metric space endowed with the metric ρ and F, G nonvoid, boundedly compact closed sets in X. Then there exist elements f . ∈ F̅ and g . ∈G ̅ such that ρ(f . , g . ) = ρ (F, G). This is false. Indeed, let Then F and G are nonvoid boundedly compact sets. But It is easy to see that the above assertionis true if we add the condition F or G is bounded Theorem 3.5. [4] Let A and B be two non-void sets of a normed linear space X. If A is weakly sequentially compact and B is convex and proximinal with respect to A, then a best proximity pair of A and B exists. For a metric space X, we assume instead that A is compact and B is proximinal with respect to A. Theorem 3.6. Let B-A be a compact set in a metric space X. Then for any two compact sets A and B in X Proof. Let {x n } ⊆A be a minimizing sequence for x, i.e.,

d(x, x n ) → d(x, A).
Since A is closed it has a limit point u ∈ A, then passing to a subsequence we may assumethatd(u, x n ) →. We then have This implies that In the same vein if {y n } ⊆B be a minimizing sequence for v in B, then it has a limit pointv in B and d(x, v) = d(x, B).
We then have Corollary 3.8 [4] Let X be a reflexive Banach space and A, B two nonvoid subsets of X. If A is bounded, weakly closed, and B is closed convex, then best proximity pair of A and B exists.
Indeed, any bounded, weakly closed subset of a reflexive space X is weakly sequentially compact. Since any closed convex subset is proximinal with respect to the total space (it follows from above theorem). Obviously, a compact set is a weakly sequentially compactset. A boundedly compact set in metric space and a closed convex set in a uniformly convex Banach space are all proximinal with respect to the total space.

Uniqueness of Mutually Nearest Points
For the uniqueness of the proximal pair, we have to impose an additional condition on X. In general, if A and B are closed and one of them is compact, the proximal pair may not exist for arbitray normed space X. However, if X is uniformly convex, then the proximal pair exists. and ∀(s, w)∈ S× Wd(S, W)≤ ||s-w|| (7) It follows from (8) and (9) that w . ∈ P W (s . ), w 1 ∈P W (s 1 ).
Below we prove Since W is strictly convex, it suffices to showthat To do this, let λ∈(0, 1). Then, by the convexity of S and W, one has (1 -λ)s . + λ s 1 ∈S.
This together with (3) and (4) It follows from (5) and (6) that for all w ∈ W.
This completes the proof. Definition 4.3. [5] Recall that a nonemptysubset A of X is called strictly convex if, for any two distinct points x, y∈ ∂A and α∈(0,1), one has that α x + (1 -α )y ∈ A.
X is called strictly convex if B(X) is strictly convex. Remark 4.4. If we only assume that one of S and W is strictly convex, the above theorem does not hold in general. However, if X is strictly convex, then the previous theorem can be improved to the following results.
Theorem 4.5. [5] Let X be strictly convex. Let one of S and W be also strictly convex and the other be quasi-convex. Then (S, W) contains at most an element. Theorem 4.6. [5] Let X be strictly convex and smooth. Let one of S and W be also strictly convex while the other be a sun. Then (S, W) contains at most an element. Remark 4.7. In Theorem 4.5 (resp. Theorem4.6), if one of S and W is not quasi-convex(resp. a sun), then the conclusion is not true in general.  Theorem 4.1. Let A, B be non-empty setsin a metric space X. If A is compact and B is closed in X and X has Pproperty, then Proof. In Theorem 3.6, we have shown that Suppose that For the uniqueness, we observe that A= {x} and B are closed sets in X. Since X has P-Property we have  For the uniqueness, we observe that B and {x} are closed sets in X. Since X has weakP-property, we have ||x-u||= ||y-v|| = d (A, B)⇒||x -y|| ≤α ||u -v||, α > 1.
Theorem 4.14. Let A and B be two sets in a metric space X and x∈A. If B is compact and X has generalized weak Pproperty then Proof. In Theorem 3.6, we have shown that For the uniqueness, we observe that B and {x} are closed sets in X. Since X has generalized weak P-property we have

Characterization of Proximal Points
The characterization of proximal points for pairs of sets, not necessarily convex is discussed in [12] by employing ''solar properties" of sets also discussed in [11]. Moreover, some duality results for the distance d(U, V) are also dealt with there.
Theorem 5.1. [12] Let X be a normed linear space whose dual X* is strictly convex. Then u ∈ U, v∈ V are proximal if and only if u is a nearest point of v in U and v is a nearest point of u in V. The characterization of proximal points are Theorem 5.2 [13] Let U, V be convex sets. Then u ̅ ∈ U, v ̅ ∈V are proximal if and only if there exists an f ∈X* such that (1) || f || = 1,

Conclusion
The mutually nearest points for two sets in a given space X has been an active topic of research for several years. In this paper, some of its more ramifications are discussed. We add the different conditions on the subsets A and B in X. We also incorporated some geometrical property of the given space X so that there exists the unique mutually nearest points. A new geometrical property called generalized weak Pproperty of a given space X is introduced and we use it for the existence and uniqueness of mutually nearest points. Finally, the characterizations of mutually nearest points are also seen. So this topic is a fertile field in the garden of many branches of mathematics.