PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 87(101) (2010), 139–142 DOI: 10.2298/PIM1001139N

A NOTE ON SUNS IN CONVEX METRIC SPACES

T. D. Narang and Sangeeta

Communicated by Stevan Pilipović

Abstract. We prove that in a convex metric space (X, d), an existence set K having a lower semi continuous metric projection is a δ-sun and in a complete M-space, a Chebyshev set K with a continuous metric projection is a γ-sun as well as almost convex.

1. Introduction

One of the most outstanding open problem of approximation theory is: Whether every Chebyshev set in a Hilbert space is convex? Several partial answers to this problem are known in the literature (see e.g.[1], [2], [3], [6], [10], and [12]) but the problem is still unsolved. While making an attempt in this direction, Effimov and Steckin [4] introduced the concept of a sun and Vlasov [13] introduced the concepts of α-, β-, γ- , δ-suns and almost convex sets in Banach spaces. These concepts were extended to convex metric spaces in [8] and some of the results proved by Vlasov [13] in Banach spaces were also proved in convex metric spaces. Continuing the study taken up in [8] and [9], we prove that in a convex metric space (X, d), an existence set K having a lower semi continuous metric projection is a δ-sun and in a complete M-space, a Chebyshev set K with a continuous metric projection is a γ-sun as well as almost convex.

2. Definitions and notations

We recall a few definitions and set up some notations. Let (X, d) be a metric space and x, y, z ∈ X. We say that z is between x and y if d(x, z)+d(z, y) = d(x, y). For any two points x, y ∈ X, the set {z ∈ X : d(x, z) + d(z, y) = d(x, y)} is called metric segment and is denoted by G[x, y]. The set G[x, y, −] denotes the half ray starting from x and passing through y, and G(x, y) ≡ G[x, y] ∖ {x, y}.

2010 Mathematics Subject Classification: 41A65.

Key words and phrases: Convex metric space, sun, proximinal set, Chebyshev set, almost convex set and lower semi continuous metric projection.

A continuous mapping W : X × X × [0, 1] → X is said to be a convex structure on X if for all x, y ∈ X and λ ∈ [0, 1]

d(u, W (x, y, λ)) ⩽ λd(u, x) + (1 − λ)d(u, y)

holds for all u ∈ X. A metric space (X, d) with a convex structure is called a convex metric space [11]. A convex metric space (X, d) is called an M-space [5] if for every two points x, y ∈ X with d(x, y) = λ, and for every r ∈ [0, λ], there exists a unique z_{r} ∈ X such that B[x, r]∩B[y, λ−r] = {z_{r}}, where B[x, r] = {y ∈ X : d(x, y) ⩽ r}. If (X, d) is a convex metric space, then for each two distinct points x, y ∈ X and for every λ (0 < λ < 1) there exists a point z ∈ X such that d(x, z) = (1−λ)d(x, y) and d(z, y) = λd(x, y). For M-spaces such a z is always unique. A normed linear space need not be an M-space. For examples of convex metric spaces and M-spaces one may refer to [5], [7] and [11].

Let V be a nonempty subset of a metric space (X, d) and x ∈ X. An element v_{0} ∈ V is called a best approximation to x if d(x, v_{0}) = dist(x, V) ≡ d_{V}(x). We denote by P_{V}(x) the set of all best approximants to x in V. The set V is said to be proximinal or an existence set if P_{V}(x) ≠ ∅ for each x ∈ X and is said to be Chebyshev if P_{V}(x) is exactly singleton for each x ∈ X. The set-valued mapping P_{V} : X → 2^{V} ≡ the set of all subsets of V, which associates with each x ∈ X, the set P_{V}(x) is called the metric projection or the best approximation map or the nearest point map. For Chebyshev sets V, the map P_{V} is single-valued.

For two topological spaces X and Y, a set-valued mapping ϕ : X → 2^{Y} is called lower semi-continuous [13] if {x ∈ X : ϕ(x) ⊂ G} is closed in X for each closed set G in Y.

A nonempty closed subset K of an M-space (X, d) is called

i. a δ-sun if for all x ∉ K there exists a sequence ⟨z_{n}⟩ for which z_{n} ≠ x, z_{n} → x and (d_{K} (z_{n}) − d_{K} (x))/d(z_{n}, x) → 1.

ii. a γ-sun if for all x ∉ K and for all R > 0 there exists z_{n} such that d_{K} (z_{n}) − d_{K} (x) → R, d(z_{n}, x) = R for all n.

iii. almost convex if for any ball V with ρ(V, K) ≡ inf{d(x, K) : x ∈ V } > 0 there exists a sufficiently large ball V′ ⊃ V with ρ(V′, K) > 0.

3. Suns in convex metric spaces

The following lemma will be used in the sequel:

Lemma 3.1. Let K be an existence set in a convex metric space (X, d) with metric projection P and suppose that x, x′, v ∈ X , are such that x′ ∈ P x and x ∈ G(x′, v). Then

(1)  0 ⩽ 1 − \frac{d(v, K) − d(x, K)}{d(v, x)} ⩽ \frac{d(x′, P v)}{d(x, K)}
Proof. Since x ∈ G(x′, v) , we can find α, 0 < α < 1 such that d(x′, x) =

(1 − α)d(x′, v) and d(x, v) = αd(x′, v) i.e., x = W (x′, v, α). Consider

d(x, x′) ⩽ d(x, P v) = d(W (x′, v, α), P v) ⩽ αd(x′, P v) + (1 − α)d(v, P v)
= αd(x′, P v) + (1 − α)d(v, K).

This implies

d(v, K) ⩾ \frac{1}{1 − α} d(x, x′) − \frac{α}{1 − α} d(x′, P v)

= d(x′, v) − \frac{d(x, v)}{d(x, x′)}d(x′, P v) = d(x′, x) + d(x, v) − \frac{d(x, v)}{d(x, x′)}d(x′, P v)

= d(x′, x) + d(x, v)[1 − \frac{d(x′, P v)}{d(x, x′)}] = d(x, K) + d(x, v)[1 − \frac{d(x′, P v)}{d(x, K)}]

i.e.,

\frac{d(v, K) − d(x, K)}{d(x, v)} ⩾ [1 − \frac{d(x′, P v)}{d(x, K)}].

This gives

(2)  \frac{d(x′, P v)}{d(x, K)} ⩾ 1 − \frac{d(v, K) − d(x, K)}{d(x, v)} .

Also d(v, K) ⩽ d(v, x) + d(x, K) implies (d(v, K) − d(x, K))/d(v, x) ⩽ 1 i.e.,

(3)  1 − \frac{d(v, K) − d(x, K)}{d(v, x)} ⩾ 0.

Therefore (2) and (3) give the desired result. □

Using the above lemma, we prove

Theorem 3.1. In a convex metric space (X, d), an existence set K having a lower semi-continuous metric projection P is a δ-sun.

Proof. Above lemma implies

(4)  0 ⩽ 1 − \frac{d(v, K) − d(x, K)}{d(v, x)} ⩽ \frac{d(x′, P v)}{d(x, K)}.

We claim that for all x′ ∈ P x, d(x′, P v) → 0 as v → x. Suppose this is not true. Then there exists x′ ∈ P x, ε > 0 and v_{n} → x such that d(x′, P v_{n}) ⩾ ε for all n. Consider the set F = {z ∈ X : d(z, x′) ⩾ ε}. Then P v_{n} ⊆ F and F is closed. By the lower semicontinuity of P, the set F_{1} = {z ∈ X : P v_{n} ⊆ F } is closed. As P v_{n} ⊆ F, v_{n} ∈ F_{1} and v_{n} → x, x ∈ F_{1} i.e., P x ⊆ F and so x′ ∈ F. This gives d(x′, x′) ⩾ ε which is absurd. Therefore for all x′ ∈ P x, d(x′, P v) → 0 as v → x and so (4) gives (d(v, K) − d(x, K))/d(v, x) → 1 i.e., K is a δ-sun. □

Corollary 3.1. In a convex metric space (X, d) a Chebyshev set K with a continuous metric projection P is a δ-sun.

Aliter Proof. As K is Chebyshev, x′ = P x. The continuity of P implies lim_{v→x} d(x′, P v) = d(x′, P x) = d(x′, x′) = 0. So (1) implies

lim_{v→x} \frac{d(v, K) − d(x, K)}{d(v, x)} = 1

i.e., K is a δ-sun. □

Since in a complete M-space, every δ-sun is a γ-sun [9], and is almost convex [8], we have:

Corollary 3.2. In a complete M-space, a Chebyshev set with a continuous metric projection is a γ-sun and almost convex.

Remark 1. For Banach spaces, the above results were proved by Vlasov [13].

References

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[3] Frank Deutsch, Best Approximation in Inner Product Spaces, Springer-Verlag, New York, 2001.

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[5] R. Khalil, Best approximation in metric spaces, Proc. Am. Math. Soc. 103 (1988), 579–586.

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[7] T. D. Narang, Sangeeta and Shavetambry Tejpal, On suns, moons and best approximation in M-spaces, Sci. Math. Jpn. 65 (2007), 1121–1124.

[8] T. D. Narang, Shavetambri Tejpal, On δ-suns, Publ. Inst. Math. 83 (2008), 99–104.

[9] T. D. Narang, Sangeeta and Shavetambri Tejpal, On various generalizations of suns, Caribb. J. Math. Comput. Sci. 15 (2008), 1–12.

[10] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York, 1970.

[11] W. Takahashi, A convexity in metric spaces and non-expansive mappings I, Kodai. Math. Sem. Rep. 22 (1970), 142–149.

[12] L. P. Vlasov, Chebyshev sets and approximatively convex sets, Math. Notes 2 (1967), 600– 605.

[13] L. P. Vlasov, Approximative properties of sets in normed linear spaces, Russian. Math. Surveys 28 (1973), 1–66.

Department of Mathematics (Received 13 04 2009) Guru Nanak Dev University Amritsar-143005 India

tdnarang1948@yahoo.co.in

Department of Mathematics Amardeep Singh Shergill Memorial College Mukandpur-144507 India

seetzz20@gmail.com