PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 83(97) (2008), 99–104 DOI: 10.2298/PIM0897099N

ON δ-SUNS

T. D. Narang and Shavetambry Tejpal

Communicated by Stevan Pilipović

Abstract. We prove that an approximatively compact Chebyshev set in an M-space is a δ-sun and a δ-sun in a complete strong M-space (or externally convex M-space) is almost convex.

The most outstanding open problem of Approximation Theory is: Whether a Chebyshev set in a Hilbert space is convex? Many attempts have been made to solve this problem. Several partial answers are known (see e.g. survey articles by Vlasov-1973 [10], Narang-1977 [6], Deutsch-1993 [3] and Balaganskii and Vlasov-1996 [1]) but in full generality, the problem is still unsolved. In order to solve the problem, Vlasov [9] introduced the concepts of δ-suns and almost convex sets in Banach spaces and proved that an approximatively compact Chebyshev set in a Banach space is a δ-sun and each δ-sun in a Banach space is almost convex. We extend these results to M-spaces [5] which are more general than Banach spaces.

To start with, we recall a few definitions. A subset K of a metric space (X, d) is said to be a δ-sun [9] if for every x ∈ X ∖ K, there is a sequence ⟨x_{n}⟩ for which x_{n} ≠ x, x_{n} → x and \frac{d(x_{n},K)−d(x,K)}{d(x_{n},x)} → 1. A closed set A in a metric space (X, d) is said to be almost convex [9] if for any closed ball B which does not intersect A, there exists a closed ball B′ ⊇ B of arbitrary large radius and which does not intersect A. For a subset K of a metric space (X, d) and x ∈ X, an element k_{0} ∈ K is said to be a best approximation to x if d(x, k_{0}) ⩽ d(x, k) for all k ∈ K i.e., d(x, k_{0}) = d(x, K) ≡ inf_{k∈K} d(x, k). The set of all such k_{0} ∈ K is denoted by P_{K}(x). The set K is said to be proximinal if P_{K}(x) ≠ ∅ for each x ∈ X and Chebyshev if P_{K}(x) is exactly singleton for each x ∈ X. The mapping p ≡ P_{K} from X into subsets of K is called the metric projection. For Chebyshev sets, p is singlevalued. The set K is said to be approximatively compact if for every x ∈ X and every sequence ⟨k_{n}⟩ in K with lim_{n→∞} d(x, k_{n}) = d(x, K) there is a subsequence ⟨k_{n_{i}}⟩ converging to an element of K.

2000 Mathematics Subject Classification: Primary 41A65; Secondary 41A50.

Key words and phrases: δ-sun, almost convex set, Chebyshev set, convex metric space, Mspace, strong M-space and externally convex space.

This research is partially supported by University Grants Commission, India (F30-238/2004(SR)).

For a metric space (X, d) and a closed interval I = [0, 1], a mapping W : X × X × I → X is said to be a convex structure on X if for all x, y ∈ X, λ ∈ I,

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

for all u ∈ X. The metric space (X, d) together with a convex structure is called a convex metric space [8]. A convex metric space (X, d) is called an M-space [5] if for every two points x, y in 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}.

An M-space (X, d) is called a strong M-space [5] if for every two points x, y in X with d(x, y) = λ and for every positive real number r ≠ λ, there exists a unique z_{r} such that S[x, r] ∩ S[y, |λ − r|] = {z_{r}} , where S[x, r] = {y ∈ X : d(x, y) = r}. A metric space (X, d) is called externally convex [5] if for all distinct points x, y such that d(x, y) = λ, and r > λ there exists a unique z of X such that d(x, y)+d(y, z) = d(x, z) = r.

Every normed linear space is a strong M-space as well as an externally convex M-space but not conversely. 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 at least one 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. For distinct points x, y of strong M-space (X, d) with d(x, y) = λ and for every r ≠ λ, there exists a unique point z of X such that d(x, y) + d(y, z) = d(x, z) = r.

We denote by G[x, y] the line segment joining x and y, i.e., G[x, y] = {z ∈ X : d(x, z) + d(z, y) = d(x, y)}; G(x, y, −) denotes the largest line segment containing G[x, y] for which x is an extreme point, i.e., the ray starting from x and passing through y; G_{1}(x, y, −) denotes the set of all those points on the ray starting from x and passing through y which do not lie between x and y.

We intend to show that approximatively compact Chebyshev sets in M-spaces are δ-suns. To develop the proof, we prove some properties of Chebyshev sets.

Lemma 1. Given a Chebyshev set K in an M-space (X, d) and metric projection x → p(x), for every x ∈ X ∖ K and x_{λ} ∈ G_{1}(p(x), x, −),

d(x_{λ}, K) ⩾ d(x, K) + d(x, x_{λ}) [1 − \frac{d(p(x), p(x_{λ}))}{d(x, p(x))}] .

Proof. Since x_{λ} ∈ G_{1}(p(x), x, −), x is between p(x) and x_{λ} so we can find some α, 0 < α < 1, such that

(*)  d(p(x), x) = (1 − α)d(p(x), x_{λ}), d(x, x_{λ}) = αd(p(x), x_{λ})

i.e., x = W (p(x), x_{λ}, α). Consider

d(x, p(x)) ⩽ d(x, p(x_{λ})) = d(W (p(x), x_{λ}, α), p(x_{λ}))
⩽ αd(p(x), p(x_{λ})) + (1 − α)d(x_{λ}, p(x_{λ})).

ON δ-SUNS 101

This implies

d(x_{λ}, p(x_{λ})) ⩾ \frac{1}{1 − α}d(x, p(x)) − \frac{α}{1 − α}d(p(x), p(x_{λ})).

Therefore

d(x_{λ}, K) = d(x_{λ}, p(x_{λ}))

⩾ \frac{1}{1 − α}d(x, p(x)) − \frac{α}{1 − α}d(p(x), p(x_{λ}))

= d(x_{λ}, p(x)) − \frac{d(x, x_{λ})}{d(p(x), x)}d(p(x), p(x_{λ})) (using (*))

= d(p(x), x) + d(x, x_{λ}) − \frac{d(x, x_{λ})}{d(x, p(x))}d(p(x), p(x_{λ})) as x ∈ [p(x), x_{λ}]

= d(x, p(x)) + d(x, x_{λ})[1 − \frac{d(p(x), p(x_{λ}))}{d(x, p(x))}].  □

Lemma 2. Given a Chebyshev set K in an M-space (X, d), if the metric projection x → p(x) is continuous on X, then

lim_{x_{λ}→ x} \frac{d(x_{λ}, p(x_{λ})) − d(x, p(x))}{d(x_{λ}, x)} = 1

for every x ∈ X ∖ K and x_{λ} ∈ G_{1}(p(x), x, −) i.e., K is a δ-sun.

Proof. We have

1 = \frac{d(x_{λ}, x)}{d(x_{λ}, x)} = \frac{d(x_{λ}, p(x)) − d(x, p(x))}{d(x_{λ}, x)}

⩾ \frac{d(x_{λ}, p(x_{λ})) − d(x, p(x))}{d(x_{λ}, x)}

⩾ 1 − \frac{d(p(x), p(x_{λ}))}{d(x, p(x))}, by Lemma 1

→ 1 as by the continuity of p, p(x_{λ}) → p(x).

The lemma is proved. □

Theorem 1. An approximatively compact Chebyshev set in an M-space is a δ-sun.

Proof. Let K be an approximatively compact Chebyshev set in an M-space (X, d) and p : X → K be the metric projection. Since the metric projection onto an approximatively compact Chebyshev set is continuous [7, p.390], p is continuous and so by Lemma 2, K is a δ-sun. □

Remark 1. For Banach spaces, this result was proved by Vlasov [9] (see also [2, p.44]).

Almost convex sets (which are very close to convex sets) and δ-suns were introduced by Vlasov [9] to solve the problem of convexity of Chebyshev sets. We now show that in complete strong M-spaces (or externally convex M-spaces), δ-suns are almost convex. For this, we shall use the Primitive Ekeland form of the Bishop– Phelps Theorem (see [4, p.167]) stated below to derive a property for a Chebyshev set in a complete strong M-space (or externally convex M-space) when the metric projection is continuous.

Primitive Ekeland Theorem. Let (X, d) be a complete metric space and ψ be a proper but extended real lower semi-continuous function on X bounded below. Then given ϵ > 0 and x_{1} ∈ X there exists an x_{0} ∈ X such that ψ(x_{0})+ϵd(x_{0}, x_{1}) ⩽ ψ(x_{1}) and ψ(x) > ψ(x_{0}) − ϵd(x_{0}, x) for all x ∈ X ∖ x_{0}.

Lemma 3. Let (X, d) be a complete strong M-space (or externally convex Mspace), K ⊆ X be a Chebyshev set with continuous metric projection x → p(x). Given x ∈ X ∖ K, r > 0 and σ > 1, there exists an x_{0} ∈ X such that

(1) d(x, K) + \frac{1}{σ} d(x, x_{0}) ⩽ d(x_{0}, K),

(2) d(y, K) < d(x_{0}, K) + \frac{1}{σ} d(y, x_{0}) for all y ≠ x_{0} and d(y, x) ⩽ r,

(3) d(x_{0}, x) = r.

Proof. Apply Primitive Ekeland Theorem to the complete metric space B[x, r] and the continuous real mapping ψ on B[x, r] defined by ψ(y) = −d(y, K). For ϵ = \frac{1}{σ}, there exists an x_{0} ∈ B[x, r] such that ψ(x_{0}) + \frac{1}{σ} d(x_{0}, x) ⩽ ψ(x) and

ψ(y) > ψ(x_{0}) − \frac{1}{σ} d(x_{0}, y) for all y ∈ B[x, r] ∖ \{x_{0}\}.

So,

d(x, K) + \frac{1}{σ} d(x_{0}, x) ⩽ d(x, K),

which proves (1), and

d(y, K) < d(x_{0}, K) + \frac{1}{σ} d(x_{0}, y) for all y ≠ x_{0} and d(y, x) ⩽ r

which proves (2).

Now, we shall prove (3). From (1), d(x, K) ⩽ d(x_{0}, K) so x_{0} ∉ K. Also x_{0} ∈ B[x, r] implies d(x, x_{0}) ⩽ r. Suppose d(x, x_{0}) < r. Take x_{0_{λ}} ∈ G_{1}(p(x_{0}), x_{0}, −), λ > 0. Then

d(x_{0_{λ}}, x) ⩽ d(x_{0_{λ}}, x_{0}) + d(x_{0}, x) < d(x_{0_{λ}}, x_{0}) + r → r as x_{0_{λ}} → x_{0}, x_{0_{λ}} ≠ x_{0}.

Therefore x_{0_{λ}} ∈ B[x, r] as x_{0_{λ}} → x_{0}, x_{0_{λ}} ≠ x_{0} i.e., for λ sufficiently small. So, from (2) we have,

\frac{1}{σ} > \frac{d(x_{0_{λ}}, K) − d(x_{0}, K)}{d(x_{0}, x_{0_{λ}})}

for sufficiently small λ. Since σ > 1,

lim_{x_{0_{λ}} →x} \frac{d(x_{0_{λ}}, K) − d(x_{0}, K)}{d(x_{0}, x_{0_{λ}})} < 1,

contradicting Lemma 2. Therefore d(x_{0}, x) = r, which proves (3). □

Theorem 2. Each δ-sun K in a complete strong M-space (or externally convex M-space) (X, d) is almost convex.

Proof. Let x ∈ X ∖ K and B[x, α] be a ball with positive distance from K. Then d(x, K) > α. Choose β > d(x, K) > α i.e., β − d(x, K) < β − α. Choose σ > 1 and r > 0 such that σ(β − d(x, K)) < r < β − α. By Lemma 3, there exists an x_{0} ∈ X such that d(x, x_{0}) = r and d(x, x_{0}) ⩽ σ(d(x_{0}, K) − d(x, K)).

Now d(x, x_{0}) = r < β − α. Also

σ(β − d(x, K)) < r = d(x, x_{0}) ⩽ σ(d(x_{0}, K) − d(x, K))

implies β − d(x, K) < d(x_{0}, K) − d(x, K) i.e., d(x_{0}, K) > β. We claim that (1) B[x_{0}, β] does not intersect K.

(2) B[x_{0}, β] ⊇ B[x, α].

Suppose B[x_{0}, β] intersects K then there exists y ∈ B[x_{0}, β]∩K i.e., d(y, x_{0}) ⩽ β and so d(x_{0}, K) ⩽ β, a contradiction. This proves (1).

Now, suppose y ∈ B[x, α]. Then d(x, y) ⩽ α. Consider

d(y, x_{0}) ⩽ d(y, x) + d(x, x_{0}) ⩽ α + r < β

i.e., y ∈ B[x_{0}, β]. This proves (2) and hence K is almost convex. □

Combining Lemma 2 and Theorem 2, we get

Theorem 3. If K is a Chebyshev set in a complete strong M-space (or externally convex M-space) (X, d) and the metric projection is continuous then K is almost convex.

Remark 2. For Banach spaces, Theorem 2 is given in [2, p.44] and Theorem 3 is given in [4, p.240].

Combining Theorems 1 and 2, we get:

Theorem 4. An approximatively compact Chebyshev set in a complete strong M-space (or externally convex M-space) is almost convex.

Acknowledgement

The authors are thankful to the referee for valuable comments leading to improvement of the paper.

References

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Department of Mathematics (Received 26 05 2005) Guru Nanak Dev University (Revised 02 11 2007) Amritsar, India

tdnarang1948@yahoo.co.in
shwetambry@gmail.com