PanXuAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 7, No. 1, 2006 pp. 41-50 Criteria of strong nearest-cross points and strong best approximation pairs Wenxi Pan and Jingshi Xu ∗ Abstract. The concept of strong nearest-cross point (strong n.c. point) is introduced, which is the generalization of strong uniqueness of best approximation from a single point. The relation connecting to localization is discussed. Some criteria of strong n.c. points are given. The strong best approximation pairs are also studied. 2000 AMS Classification: 41A50, 41A28, 41A65. Keywords: strong nearest-cross point, local strong nearest-cross point, strong best approximation pair. 1. Introduction In [6], [5], [9] the first author of the paper studied the nearest cross points (in short, n.c. points) of two subsets of a normed space. More precisely, let G and F be two disjoint subsets of a normed space X. A point y0 ∈ G is called a n.c. point of G to F if ρ(y0, F ) = ρ(G, F ), where ρ(G, F ) = infy∈G,x∈F ρ(y, x), ρ(y, x) = ‖x − y‖ is the norm of x − y in the space X. Moreover, if x0 ∈ F satisfies ρ(x0, y0) = ρ(F, G), we say that (x0, y0) is a best approximation pair of F and G. For details, one can see [4]. Obviously, if (x0, y0) is a best approximation pair of F and G, then y0 is a n.c. point of G to F , and x0 is the best approximation of y0 from F . The analogous result for x0 also holds. However, the inverse is not true. If both n.c. points of F to G and G to F exist, a best approximation pair of F and G may not exist. But if n.c. points of G to F exist and F is a proximal set, then the best approximation pair of F and G exists. In [5], the author discussed the uniqueness of n.c. points (if it exists) and obtained that the n.c. point of G to F is unique if G is strict convex and F is convex. In this paper, we shall discuss a property which is stronger than being a n.c. point, which we will call a strong n.c. point. A strong n.c. point is ∗The corresponding author Jingshi Xu was supported by the NNSF (No. 60474070) of China 42 W. Pan and J. Xu the generalization of strong best approximation in a single best approximation problem. For strong best approximation, one can see [3],[7],[2] in detail. The organization of the paper is as follows. In Section 2, we will give the definition of strong n.c. point. In Section 3, we shall discuss the criteria of strong nearest cross point. In Section 4, we shall discuss strong n.c. points and strong best approximation pairs by way of the concept of cusp. And we shall give more examples about the relation between strong best approximation pairs and strong n.c. points. Finally, we declare that we will work in complex norm spaces in this paper and use the following notation. Let X be a normed space. Denote by X∗ the dual space of X. For a complex number u, we shall write Re u to denote the real part of u. If F denotes a subset of a normed space X, then ‖F ‖ = supx∈F ‖x‖. 2. Definition of strong n.c. point Definition 2.1. Let F and G be two disjoint sets, y0 ∈ G and a constant r, 0 < r < 1. If the condition (1) ρ(y, F ) − ρ(y0, F ) ≥ rρ(y, y0) holds for every y ∈ G, then y0 is called a strong n.c. point of G to F . Notice that a strong nearest point is a n.c. point. In fact, since ρ(y, F ) − ρ(y0, F ) ≥ 0 for every y ∈ G, we have ρ(G, F ) = ρ(y0, F ). Then, for every y ∈ G, y 6= y0, ρ(y, F ) > ρ(y0, F ), thus the n.c. point is unique. In Definition 2.1, the constant r, r < 1 holds automatically because |ρ(y, F ) − ρ(y′, F )| ≤ ρ(y, y′) always holds. In fact, to say y0 is a strong n.c. point, it suffices to remark that it exists a sufficiently small r, such that (1) holds. If F is a singleton x0, y0 is a strong n.c. point of G to F, then y0 is the strong (unique) best approximation of x0 from G; see [3], [7], [2]. Definition 2.2. Consider F, G, r, y0 as in Definition 2.1. If (1) holds only for y ∈ V0 ∩ G, V0 a neighborhood of y0, then we say y0 is a local strong n.c. point of G to F. Obviously, if y0 is a strong n.c. point then y0 is a local strong n.c. point, but the converse does not hold in general, as the following example shows. Example 2.3. In the Euclidean space R2, let F = {(ξ, η) : (ξ−2)2+η2 = 1, ξ ≥ 2}, G = {(ξ, η) : ξ2 + η2 = 1, ξ ≤ 0}. If y0 = (0, 1), then y0 is a local strong n.c. point of G toF , since for every y ∈ G, y be near y0, ρ(y, F ) is equivalent to ρ(y, x0) = 2 + ρ(y, y0). x0 = (2, 1). So ρ(y, F ) − ρ(y0, F ) is equivalent to ρ(y, y0). But y0 is not a strong n.c. point. In fact, choose y ′ suffice to (0, −1), then ρ(y0, y ′) is to 2, ρ(y0, F ) = 2, thus ρ(y ′, F ) − ρ(y0, F ) converges to 0, so (1) does not hold. Moreover, (2,-1) is a n.c. point of F to G, (0,1) is a nearest cross point of G to F , and (2,-1), (0,1) are not strong n.c. points. In the above example, F and G are not convex sets, but under convexity, we shall have different results. To state our results, we need the following lemma, which is well known. Criteria of strong nearest-cross points 43 Lemma 2.4. Let (X, ρ) be a metric space and let F ⊂ X. Then the function ρ(·, F ) is uniformly continuous. Moreover if F is a convex set, then ρ(·, F ) is a convex function. Theorem 2.5. Let F , G be convex sets, and let y0 ∈ G. y0 is a strong nearest cross point of G to F if and only if y0 is a local strong n.c. point of G to F . Proof. As above stated, we only need to show that if y0 is a local strong n.c. point, then y0 is a strong n.c. point. Let V be a neighborhood of y0, where y0 is a strong n.c. point of G ∩ V to F . If y0 is not a strong n.c. point of G to F , then for every rn → 0, there exist yn ∈ G, n = 1, 2, . . . , such that ρ(yn, F ) − ρ(y0, F ) < rn‖yn − y0‖. In the segment [yn, y0], pick zn = λnyn + (1 − λn)y0, 0 < λn < 1, λn → 0, such that for n sufficiently large, zn ∈ G ∩ V. By Lemma 2.4, ρ(y, F ) is a convex function. Thus, ρ(zn, F ) ≤ λnρ(yn, F ) + (1 − λn)ρ(y0, F ) < λnrn‖yn − y0‖ + λnρ(y0, F ) + (1 − λn)ρ(y0, F ) = λnrn‖yn − y0‖ + ρ(y0, F ). So ρ(zn, F ) − ρ(y0, F ) < λnrn‖yn − y0‖. Since ‖yn − y0‖ = 1/λn‖zn − y0‖, ρ(zn, F ) − ρ(y0, F ) ≤ rn‖zn − y0‖, zn ∈ G ∩ V. This contradicts the definition of a local strong n.c. point. This completes the proof. � In Theorem 2.5 we suppose F and G are convex sets. If one of them is not convex, then the result does not hold. Example 2.6. In the Euclidean space R2, let G = {(ξ, η) : ξ2 + η2 = 1, ξ ≤ 0}. Notice that G is not convex and F = {(2, 0)} is a singleton. Then y0 = (0, 1) is a local strong n.c. point of G to F, but y0 is not a strong n.c. point of G to F . In fact, pick y = (0, −1). Then ρ(y, F ) − ρ(y0, F ) = √ 5 − √ 5 = 0, but ‖y − y0‖ = 2. Example 2.7. In the Euclidean space R2, let G = {(ξ, η) : −3 ≤ ξ ≤ 2, η = 0} be a convex set, indeed, a segment, and F = {(ξ, η) : ξ2 + η2 = 25} a non- convex set. Then y0 = (2, 0) is a local strong n.c. point, but y0 is not a strong n.c. point, even if it is not a n.c. point. In fact, (-3,0) is a strong n.c. point. 3. Kolmogorov type and differential type criteria of strong n.c. points In [6], the author gave sufficient and necessary conditions for a point to be a n.c. point by means of linear functions and differentials. Following the same idea we first obtain a sufficient condition for strong n.c. points. Theorem 3.1. Let F , G be disjoint sets, y0 ∈ G, and r a constant, 0 < r < 1. If y0 satisfies one of the following conditions, then y0 is a strong n.c. point of G to F . (i) for every ǫ > 0, there exists f ǫ ∈ X∗, ‖f ǫ‖ = 1, such that inf x∈F Re f ǫ(x−y) = ρ(y0, F ) and for every y ∈ G, Re f ǫ(y0 − y) + ǫ ≥ r‖y0 − y‖ holds. 44 W. Pan and J. Xu (ii) for every ǫ > 0, and y ∈ G, there exists f y,ǫ ∈ X∗, ‖f y,ǫ‖ = 1, such that inf x∈F Re f y,ǫ(x − y0) ≥ ρ(y0, F ), and Re f y,ǫ(y0 − y) + ǫ ≥ r‖y0 − y‖. Proof. Obviously, (i) implies (ii). So we only need to show (ii). For this, given ǫ > 0, y ∈ G, and f y,ǫ as in (ii), we have r‖y0−y‖ ≤ Re f y,ǫ(y0−y)+ǫ = Re f y,ǫ(y0−x)+Re f y,ǫ(x−y)+ǫ ≤ ‖x−y‖−ρ(y0, F )+ǫ. If in the right side of the above inequality, we take the infimum over x ∈ F, we have r‖y0 − y‖ ≤ ρ(y, F ) − ρ(y0, F ). Thus y0 is the strong n.c. point of G to F . This completes the proof. � One may ask immediately if either (i) or (ii) are necessary conditions. To answer this question, we begin with directional derivatives of the distance func- tion ρ(y, F ) (for details one can see [6]). From Lemma 2.4, we know ρ(y, F ) is a convex function. Furthermore, ρ′+(y, h, F ) = lim t→0+ ρ(y + th, F ) − ρ(y, F ) t exists for h 6= 0 ∈ X, and ρ′+(y, h, F ) is a subadditive homogenous function in the variable h. Theorem 2.2 in [6] says that whenever y0 is a n.c. point of G to F, ρ′+(y0, y − y0) ≥ 0 for all y ∈ G. In the following, we shall obtain a necessary condition for strong n.c. points. Theorem 3.2. Let F , G be two disjoint convex sets of X. If y0 ∈ G is a strong n.c. point of G to F, then (Ds) ρ ′ +(y0, y − y0, F ) ≥ r‖y − y0‖ for all y ∈ G, r > 0. Proof. From Definition 2.1, ρ(y, F ) − ρ(y0, F ) ≥ r‖y − y0‖. Put h = y − y0, ρ(y0 + th, F ) − ρ(y0, F ) t ≥ r‖th‖ t = r‖h‖. Set t towards to 0 from right of 0, then ρ′+(y0, h, F ) ≥ r‖y − y0‖. Note that t > 0, t is sufficiently small and y0 + th ∈ G, since G is convex. This completes the proof. � We shall consider whether condition (Ds) is sufficient. Some lemmas are required. To state them we give first some notation. Γ = {ϕ ∈ X∗ : ‖ϕ‖ = 1, Re ϕ(u) ≤ ϕ(y0), for u ∈ H}, here, H = {u : ρ(u, F ) ≤ ρ(F, G)}. NF = {f ∈ X∗ : ‖f‖ = 1, inf x∈F Re f (x) = inf u∈F ‖u‖}. The following lemma is Lemma 2.3 and Lemma 2.4 in [6]. Lemma 3.3. For every h 6= 0 ∈ X, supϕ∈Γ Re ϕ(h) ϕ(y0) = ρ ′ + (y0,h,F ) ρ(y0,F ) , and −Γ = NF −y0. Criteria of strong nearest-cross points 45 Theorem 3.4. Let F be a subspace of X, G a convex subset of X, F ∩ G = ∅ and 0 < r < 1. If (Ds) holds, then the following condition holds (Ks) for every ǫ > 0, and every y ∈ G, there exists f0 (depend on ǫ, y) ∈ X∗ with ‖f0‖ = 1, such that Re f0(y0 − y) + ǫ ≥ r‖y0 − y‖. Proof. By the condition (Ds), Theorem 3.1 and Lemma 3.3, we have sup f∈NF −y0 Re f (y0 − y) Re f (y0) ≥ r‖y − y0‖ ρ(y0, F ) . So for every ǫ, y ∈ G, there exists f0 ∈ NF −y0, such that Re f0(y0−y) Re f0(y0) + ǫ ρ(y0,F ) ≥ r‖y−y0‖ ρ(y0,F ) . Since 0 ∈ F, the definition of NF −y0 , Re f0(y0) ≥ ρ(y0). This implies that Re f0(y0 −y)+ǫ ≥ r‖y0 −y‖, so (Ks) holds. This completes the proof. � Finally, we give a condition (Bs) which is equivalent to (Ks) for general disjoint sets F , G. Before stating it, we require a notation. Let F be a subset of the space X. Denote QF = {u : Re φ(u) ≤ ‖F ‖, for all φ ∈ NF }. Notice that QF is a cone type set including F. For if z ∈ QF , x ∈ F, z′ = x + t(z − x), t > 0, then z′ ∈ QF . Because for every φ ∈ NF , Re φ(z) ≤ ‖F ‖, Re φ(z′) = Re φ(x) + tRe φ(z −x) = (1−t)Re φ(x) + tφ(z) ≤ (1−t)‖F ‖+ t‖F ‖ = ‖F ‖. Specially, if F is a singleton x0, then QF is a cone including ball B(0, ‖x0‖); see [3]. Theorem 3.5. Let F , G be two disjoint sets, then (Ks) is equivalent to (Bs) QF −y0 ∩ cone(y0 − G) is bounded, cone(E) denotes the cone closure of E. Proof. Suppose (Ks) holds, for every y ∈ G. Then sup y∈NF −y0 Re f (y0 − y) ≥ r‖y0 − y‖, 0 < r < 1. We should conclude that QF −y0 ∩ cone(y0 − G) ⊂ B(0, ‖F − y0‖/r). If this is not true, there exists t > 0, and some y ∈ G such that t(y0 − y) ∈ QF −y0 , ‖t(y0 − y)‖ > 1/r‖F − y0‖. From the definition of QF −y0 , for every f ∈ NF −y0 , |tRe f (y0 − y)| ≤ ‖F − y0‖. If we take the supremum over all f ∈ NF −y0 , then ‖F − y0‖ ≥ tr‖y0 − y‖ > ‖F − y0‖, which leads us to a contradiction. Now if (Bs) holds, from the above statement, there exists a sufficient large number α, such that QF −y0 ∩cone(y0−G) ⊂ intB(0, α), α > 0. So for every y ∈ G, y0 −y ∈ y0 −G, and y0−y‖y0−y‖ ∈ cone(y0 −G). But since α y0−y ‖y0−y‖ /∈ intB(0, α), then α y0−y‖y0−y‖ /∈ QF −y0 . There exists f0 ∈ NF −y0 , such that Re f0( α(y0−y) ‖y0−y‖ ) > ‖F −y0‖. It means that Re f0(y0−y) > 1α ‖F −y0‖‖y0−y‖. If we put r = ‖F −y0‖ α and we take α large enough such that 0 < r < 1, then (Ks) holds. � 46 W. Pan and J. Xu 4. Cusp and strong best approximation pairs In this section, we shall discuss the case when either strong nearest cross point or strong best approximation pair involve a cusp. In the end of this section, we shall give three examples of strong best approximation pairs. Let us begin with the definition of a cusp. Definition 4.1. Let G be a nonempty subset of X, and let ∂G be the boundary of G. Given y0 ∈ G ∩ ∂G, a point y0 is called a cusp of G if there exists a hyperplane P supporting G at y0, and ρ(y,P ) (y,y0) > σ > 0 holds for every y ∈ G, where σ is a constant. Obviously, every cusp is a strongly exposed point. We say that y0 is a strongly exposed point of G if there exists a hyperplane P supporting G at y0, x ∈ P, f (x) = c and such that if for every arbitrary ǫ > 0, there exists δ > 0, such that |f (y) − f (y0)| < δ for y ∈ G, then ρ(y, y0) < ǫ. In fact, ρ(y, P ) = |f (y)−c| ‖f‖ . Without loss of generality, we assume ‖f‖ = 1, then ρ(y, P ) = |f (y) − f (y0)|. Since y0 is a cusp of G, ρ(y,P )ρ(y,y0) > σ, ρ(y, y0) < |f (y) − f (y0)|/σ for y ∈ G. For exposed points and strongly exposed points, one can see [1], [8] and the references there in. Let F , G be two nonempty sets with ρ(F, G) > 0. We say that two hy- perplanes P , Q regular separate F and G, if P , Q are parallel, F and G are in two outer sides of P and Q, and ρ(P, Q) = ρ(F, G) = ρ(F, P ) = ρ(G, Q). Furthermore, if y0 ∈ G is such that ρ(y0, F ) = ρ(F, G), and ρ(y,P )ρ(y,y0) > δ > 0, we say that y0 is cusp of G to F . Obviously, if y0 is a cusp of G to F, then y0 is a cusp of G. Theorem 4.2. Let F , G be two disjoint convex sets and let y0 ∈ G. If y0 is a cusp of G to F , then y0 is a strong n.c. point of G to F . Proof. From the definition of cusp of G to F , there exist hyperplanes P , Q sep- arating F , G such that ρ(y,P ) ρ(y,y0) > δ for every y ∈ G. Then, ρ(y, F ) − ρ(y0, F ) ≥ ρ(y, Q) − ρ(y0, F ). Note that since P is parallel to Q, then ρ(y, Q) = ρ(y, P ) + ρ(P, Q) and ρ(P, Q) = ρ(y0, F ). Thus ρ(y, F ) − ρ(y0, F ) = ρ(y, P ) > δρ(y, y0). This means that y0 is a strong n.c. point. � Let h be the Hausdorff metric h(F0, F1) = max{△(F0, F1), △(F1, F0)}, where △(F0, F1) = sup x∈F0 inf x′∈F1 ‖x − x′‖. We have Theorem 4.3. (Freud type proposition) Suppose y0 is a strong n.c. point of G to F. If y1 is a strong n.c. point of G to F1, then ‖y − y0‖ < 2/rh(F0, F1). Proof. According to Definition 2.1, there exists 0 < r < 1, such that r‖y1 − y0‖ ≤ △(F, G) + ρ(y1, F ) − ρ(y0, F ). It is easy to see that ρ(y, B) − ρ(y, A) ≤ (A, B) holds for every y. Thus r‖y1 − y0‖ ≤ △(F1, F0) + ρ(y1, F1) − ρ(y0, F0) ≤ △(F1, F0) + ρ(y0, F1) − ρ(y0, F0) ≤ △(F1, F0) + △(F0, F1) ≤ 2h(F0, F1), Criteria of strong nearest-cross points 47 (to obtain the second inequality, we used that y1 is a strong n.c. point of G to F ). This completes the proof. � In smooth normed spaces, F is a singleton and G is a normed subspace, then Theorem 4.3 is the result of Wulbert [3, page 95]. Theorem 4.4. Let F , G be two disjoint sets with F convex and G a linear subspace and y0 ∈ G. If ρ(y, F ) is Gateaux differential at y0, then y0 is not a strong n.c. point of G to F . Proof. Suppose y0 is a strong n.c. point. By the Gateaux differentiable of ρ(y, F ), we have ρ′ (y0, h, F ) + ρ ′ +(y0, −h, F ) = 0, for h 6= 0. By Theorem 3.2, ρ′+(y0, y − y0, F ) ≥ r‖y − y0‖ holds for all y 6= y0, y ∈ G, where 0 < r < 1. If y − y0 is either h or −h, we have 0 ≥ 2r‖y − y0‖, which is a contradiction. The proof is complete. � Definition 4.5. Let F , G be two disjoint sets, x0 ∈ F, and y0 ∈ G. We say that (x0, y0) is a strong best approximation pair of F and G if there exist positive constants r, r′ such that ρ(y, y) − ρ(x0, y0) ≥ r‖x − x0‖ + r′‖y − y0‖ for all x ∈ F, y ∈ G. Obviously, a strong best approximation pair of F and G is a best approxi- mation pair of F and G; for best approximation pairs one can see [5] in detail. In the following, we shall discuss the connection between strong best approxi- mation pairs and strong n.c. points. Theorem 4.6. If F , G are two disjoint sets, then (x0, y0) is the strong best approximation pair of F and G, if and only if, y0 is a strong n.c. point of G to F , x0 is the strong n.c. point of F to G, and (x0, y0) is a best approximation pair of F and G. In this case, it is unique. Proof. If (x0, y0) is the strong best approximation pair of F and G, by Defini- tion 4.5, ρ(x0, y0) = ρ(F, G) = ρ(y0, G) and ρ(x, y) − ρ(y0, F ) ≤ r′‖y − y0‖ for all x ∈ F. If we take the infimum over all x ∈ F, we have ρ(y, F ) − ρ(y,F ) ≥ r′‖y − y0‖. Thus y0 is the strong n.c. point of G to F . Similarly, x0 is a strong n.c. point of F to G. Conversely, since (x0, y0) is a best approximation pair of F and G, then ρ(x0, y0) = ρ(F, G) = ρ(y0, F ) = ρ(x0, G). Note that ρ(x, y) ≥ ρ(y, F ) and y0 is a strong n.c. point of G from F . So, ρ(x, y)−ρ(x0, y0) ≥ ρ(y, F )−ρ(y0, F ) ≥ r′‖y − y0‖. Similarly, ρ(x, y)− ρ(x0, y0) ≥ r‖x− x0‖. Thus, ρ(x, y)− ρ(x0, y0) ≥ r/2‖x − x0‖ + r′/2‖y − y0‖. Therefore (x0, y0) is a strong best approximation pair. This completes the proof. � Theorem 4.7. Let F , G be two disjoint sets, ρ(F, G) > 0, and (x0, y0) a best approximation pair of F and G. If y0 is a cusp of G to F , and x0 is a cusp of F to G, then (x0, y0) is a strong best approximation pair. Proof. By the definition, y0 is a cusp of G to F , and there exist parallel hy- perplanes separating P , Q, such that F and G are in the outer side of P and Q, y0 ∈ P, and ρ(P, Q) = ρ(F, G) = ρ(P, F ) = ρ(Q, G) = ρ(y0, F ). Obviously, 48 W. Pan and J. Xu ρ(x, y) ≥ ρ(y, Q) for all x ∈ F, y ∈ G. Since (x0, y0) is a best approximation pair of F and G, ρ(x0, y0) = ρ(y0, F ). So ρ(x, y)−ρ(x0, y0) ≥ ρ(y, Q)−ρ(y0, F ). Note that since ρ(y0, F ) = ρ(P, Q), and ρ(y, Q) = ρ(y, P ) + ρ(P, Q), then ρ(x, y) − ρ(x0, y0) > ρ(y, P ) > σ′‖y − y0‖. Thus ρ(x, y) − ρ(x0, y0) > σ/2‖x − x0‖ + σ′/2‖y − y0‖. This completes the proof. � Finally, we shall give three examples. Example 4.8. Denote C[0,1] be all continuous function f on [0,1] with norm ‖f‖ = maxt∈[0,1] |f (t)|. In C[0,1], let F = {µt : −∞ < µ < ∞}, G = {λt2 :√ 2 + 1 ≤ λ ≤ 5}. We consider n.c. points, best approximation pairs, strong n.c. points and strong best approximation pairs between F and G. Denote x(t) = µt, y(t) = λt2. Then ρ(x, y) = ‖µt − λt2‖ = { λ − µ, for µ/λ ≤ √ 8 − 2 µ2/4λ, for µ/lz ≥ √ 8 − 2. First we compute inf −∞<µ<∞ ‖µt−λt2‖. For fixed λ, ‖µt−λt2‖ takes its infimum at µ = µλ. By the representation of ‖µt − λt2‖, µλ satisfies λ − µλ = µ2λ/4λ, so µλ = ( √ 8 − 2)λ. Thus ρ(F, G) = inf√2+1≤λ≤5 ‖λt2 − ( √ 8 − 2)λt‖ = ( √ 2 + 1)‖t2 − ( √ 8 − 2)t‖ = ( √ 2 + 1)(3 − √ 8) = √ 2 − 1. From above we obtain that ‖x0 − y0‖ = ρ(F, G), when x0(t) = µ0t, y0(t) = λ0, λ0 = √ 2 + 1, µ0 = 2. We declare that y0(t) is a strong n.c. point of G to F, since for every y(t) = λt2 ∈ F, ρ(y, F ) = inf −∞<µ<∞ ‖λt2 − µt‖ = ‖λt2 − µλt‖ = λ‖t2 − ( √ 8 − 2)t‖ = (3 − √ 8)λ. Therefore, ρ(y0, F ) = ρ0 = (3 − √ 8)λ0, ρ(y, F ) − ρ(y0, F ) = (3 − √ 8)(λ − λ0) = r′‖y − y0‖, r′ = 3 − √ 8. Similarly, we have that x0 is a strong n.c. point of F to G, since for every x(t) = µt, ρ(x, G) = inf√ 2+1≤λ≤5 ‖λt2 − µt‖ = inf√ 2+1≤λ≤5 { λ − µ, forλ ≥ µ/ √ 8 − 2 µ2/4λ, forλ ≤ µ/ √ 8 − 2. So, ρ(x, G) takes the infimum at λµ = √ 2 + 1/2µ, and ρ(x, G) = λµ − µ =√ 2/2µ. Thus, ρ(x, G) − ρ)x0, G) = √ 2 + 1/2µ − √ 2 − 1 = r‖x − x0‖, where r = √ 2 − 1/2. By Theorem 3.4, we obtain that (x0, y0) is the strong best approximation pair of F and G. The following example shows that a n.c. point always exists, but strong n.c. points can fail to exist. Example 4.9. Denote ℓ21 = {(ξ, η) : ξ, η ∈ R, ‖(ξ, η)‖ = |ξ| + |η|}. In ℓ21 space, let F = {(ξ, η) : ξ = η}, G = {(ξ, η) : η = 0, 2 ≤ ξ ≤ 3}. It is easy to see that y0 = (2, 0) is a n.c. point of G to F . But the best approximation of (2,0) Criteria of strong nearest-cross points 49 from F is not unique. Thus the nearest cross points of G to F is not unique. Therefore a strong n.c. point of G to F does not exist . At the end, we shall give an example, which shows that it is possible to find best approximation pairs which are not strong best approximation pairs. Example 4.10. In the Euclidean R2 space, set F = {(ξ, η) : ξ2 + η2 ≤ 1}, G = {(ξ, η) : ξ ≥ 2, −ξ + 2 ≥ η ≥ ξ − 2}. It is easy to see that x0 = (1, 0), y0 = (2, 0) is the unique best approximation pair of F and G. Denote an arbitrary point of F as x = (cos θ, sin θ), 0 ≤ θ < 2π. Put y = (2 + δ, δ) ∈ G, δ > 0. Then ρ0 = ρ(x0, y0) = 1, ρ 2 = ρ2(x, y) = (2 + δ − cos θ)2 + (δ − sin θ)2. Setting θ → 0, δ → 0, then ρ2 − ρ20 is asymptotic to 2(ρ − ρ0). ‖x − x0‖ = | sin θ|, ‖y − y0‖ = √ 2δ. Since δ ρ2−ρ2 0 , and θ ρ2−ρ2 0 are not bounded, then ‖y−y0‖ ρ−ρ0 , and ‖x−x0‖ ρ−ρ0 are also not bounded. This means that (x0, y0) is not a strong best approximation pair. However, y0 is a strong n.c. point of G to F . Acknowledgements. The authors would like to give their deep gratitude to the referee for his careful reading of the manuscript and his suggestions which made this article more readable. References [1] P. Beneker and J. Wiegerinck, Strongly exposed points in uniform algebras, Proc. Amer. Math. Soc. 127 (1999), 1567-1570. [2] D. Braess, Nonlinear Approximation Theory, Springer-Verlag, Berlin, Heidelberg, 1986. [3] D. F. Mah, Strong uniqueness in nonlinear approximation, J. Approx. Theory 41 (1984), 91-99. [4] W. Pan, The distance between sets and the properties of best approximation pairs, J. of Math. (Chinese), 14 (1994), 491-497. [5] W. Pan, On the existence and uniqueness of proximity pairs and nearest-cross points, J. Math. Research and exposition (Chinese), 15 (1995), 237-243. [6] W. Pan, Characterization of nearest-cross points in the problem of distance of two convex sets, J. Jinan Univ. (Natural Science) (Chinese), 20(3) (1999), 1-7. [7] P. L. Papini, Approximation and strong approximation in normed spaces via tangent function, J. Approx. Theory 22 (1978), 111-118. [8] A. A. Tolstonogov, Strongly exposed points of decomposable sets in spaces of Bochner integrable functions, Mathematical Notes 71 (2002), 267-275. [9] Q. Wang, W. Pan, New concept of normal separation of two sets and its properties, Acta Sci. Nat. Univ. Sunyatseni (Chinese), 39(6) (2000), 20-25. 50 W. Pan and J. Xu Received March 2004 Accepted December 2005 Wenxi Pan Department of Mathematics, Jinan University, 510632 Guangdong, China. Jingshi Xu (jshixu@yahoo.com.cn) Department of Mathematics, Hunan Normal University, 510632 Hunan, China