Let C be a closed bounded convex subset of a real Banach space X with 0 as its interior and pC the Minkowski functional generated by the set C. For a nonempty set G in X and x∈X, g0∈G is called the generalized best approximation to x from G if pC(g0−x)≤pC(g−x) for all g∈G. In this paper, we will give a distance formula under pC from a point to a closed hyperplane H(x∗,α) in X determined by a nonzero continuous linear functional x∗ in X and a real number α, a representation of the generalized metric projection onto H(x∗,α), and investigate the continuity of this generalized metric projection, extending corresponding results for the case of norm.

1. Introduction

Throughout this paper, (X,∥·∥) is a real Banach space with the closed unit ball B(X), and X* is its the topological dual. For a nonempty subset A of X, as usual, int A and bd A stand for the interior and the boundary of A, respectively. Let C be a bounded closed convex subset of X with 0∈ int C. Recall that the Minkowski function pC:X→ℝ with respect to the set C is defined by
(1)pC(x)∶=inf{t>0:x∈tC},∀x∈X.
Let G be a nonempty subset of X and x∈X. If there exists g0∈G such that
(2)pC(g0-x)=τC(x,G),
where
(3)τC(x,G)∶=inf{pC(g-x):g∈G}
is the distance from the point x to the set G, then following [1] g0 is called the generalized best approximation to x from G. The set of all generalized best approximations to x from G is denoted by PGC(x); that is,
(4)PGC(x)={g0∈G:pC(g0-x)=τC(x,G)},
which is called the generalized metric projection onto G.

When pC is the norm of X, the generalized best approximation is reduced to the classical best approximation, which has been studied deeply and extensively since the late 1950s; see [2–4] and references therein. Thus, natural problems are that whether we can extend results in the classical approximation theory to the setting of the generalized approximation. In this direction, some meaning results, such as existences, characterizations, and well-posedness of this kind of approximation, have been established recently; see [1, 5–7]. In this paper, we will consider the problem of representation of generalized metric projection PH(x*,α)C onto a closed hyperplane H(x*,α) and one of continuity of PH(x*,α)C, where 0≠x*∈X*, α∈R, and
(5)H(x*,α)={x∈X:x*(x)=α}.
When pC is the norm of X or, equivalently, C is the closed unit ball B(X) of X, this problem has been studied by a few authors; see [8–12]. In particular, when X is reflexive, Wang and Yu have given in [10] the representation of PH(x*,α)B(X), which was further extended by Ni in [8] to the case of nonreflexive Banach spaces. When X is nearly strictly convex, Wang has shown in [11] that PH(x*,α)B(X) is norm-to-weak upper semicontinuous on X, while, when X is arbitrary Banach space, Zhang and Shi have given in [12] the pointwise continuity of PH(x*,α)B(X) under an additional condition.

It should be noted that, when one uses a nonnegative convex function φ on the Euclidean space Rn satisfying φ(0)=0 and φ(λx)=λφ(x) for all x∈Rn and λ≥0 as a metric on Rn (i.e., the distance from a point x to a subset G of Rn is defined as d(x,G)=infg∈Gφ(g-x)), Ferreia and Nemeth have investigated in [13] the problem of the best approximation in Rn and, in particular, given some properties of corresponding metric projections on a hyperplane in Rn.

The organization of the present paper is as follows. In Section 2, we define the notions of near strict convexity and weak near strict convexity for the underlying set C, which are, respectively, natural extensions of corresponding notions in norm context, and provide an example of a real Banach space X for which B(X) is weakly nearly strictly convex but not nearly strictly convex. In Section 3, under pC, we give a distance formula from a point in X to a hyperplane H(x*,α) in X and a representation of the generalized metric projection PH(x*,α)C and consider the continuity of PH(x*,α)C. Results obtained in the present paper extend classical Ascoli Theorem (i.e., the distance formula under the case of norm from a point to a closed hyperplane in a Banach space) and main results in [8, 10–12] from the setting of norm to that of the Minkowski functional.

2. Preliminaries and an Example

Recall that (X,∥·∥) is a real Banach space with the topological dual X*, C is be a closed bounded convex subset of X with 0∈intC, and pC is the Minkowski function given by (1). Define the polar C∘ of the set C by
(6)C∘∶={x*∈X*:x*(x)≤1,∀x∈C}.
Then C∘ is a nonempty weakly* compact convex subset of X* with 0∈ int C∘.

We first list some useful properties of the Minkowski function pC which can be proved easily by the definition.

Proposition 1.

Let x,y∈X and x*∈X*. Then

pC(x)≥0 and pC(x)=0⇔x=0;

x∈C⇔pC(x)≤1 and x∈bdC⇔pC(x)=1;

p(x+y)≤p(x)+p(y) and pC(tx)=tpC(x) for each t≥0;

pC(x)=supy*∈C∘y*(x) and pC∘(x*)=supx∈Cx*(x);

x*(x)≤pC(x)pC∘(x*);

there exist positive numbers m1 and m2 such that (7)m1∥x∥≤pC(x)≤m2∥x∥.

We then give the following definitions which will be used in the rest of this paper.

Definition 2.

Let T be a set-valued mapping from X into 2X, where 2X is the set of all subsets of X.

Let x∈X with T(x)≠∅. Then T is said to be norm-to-norm (resp., norm-to-weak) upper semicontinuous at x if, for each open set (resp., weakly open set) W⊇T(x) there exists an open neighborhood V of x such that T(y)⊆W whenever y∈V.

T is said to be norm-to-norm (resp., norm-to-weak) upper semicontinuous on X if, for each x∈X, T(x)≠∅ and T is norm-to-norm (resp., norm-to-weak) upper semicontinuous at x.

T is said to be norm-to-norm continuous on X if, for each x∈X, T(x) is single valued and T is norm-to-norm upper semicontinuous at x.

Definition 3.

The set C is said to be strictly convex (resp., nearly strictly convex and weakly nearly strictly convex) if each convex subset of bd C is a singleton (resp., relatively compact and relatively weakly compact).

Clearly, the notions of near strict convexity and weak near strict convexity for the set C are extensions of corresponding notions for the unit ball B(X), which were, respectively, posed by Banaś in [14] and by Wang in [11]. In the following we will provide an example to show that the near strict convexity for B(X) is strictly stronger than the weak near strict convexity for B(X).

Example 4.

Let X=l2 be the space of all square convergent real sequences, endowed with the norm by
(8)∥x∥0=max{∥x∥22,∥x∥∞},∀x∈X,
where ∥·∥2 and ∥·∥∞ are the l2-norm and the supremum norm on X, respectively. Then ∥·∥0 is equivalent to the l2-norm on X because
(9)∥x∥22≤∥x∥0≤∥x∥2,∀x∈X.
Hence, (X,∥·∥0) is reflexive. It implies that each convex subset of bd B(X) is relatively weakly compact, and consequently B(X) is weakly nearly strictly convex. Below we show that B(X) is not nearly strictly convex. To this end, let {en} be the natural basis of l2, where the nth coordinate of en is 1 and the other coordinates are 0. Furthermore, let x1=e1 and xn=e1+en for each n≥2. We claim that co{xn}n≥1⊆bd B(X). Indeed, let y∈co{xn}n≥1. Then there exist a positive integer n and a sequence {λi}i=1n with {λi}i=1n⊆[0,1] satisfying ∑i=1nλi=1 such that y=∑i=1nλixi. Since y=e1+∑i=2nλiei, one has that ∥y∥∞=1 and
(10)∥y∥2=(1+∑i=2nλi2)1/2≤(1+(∑i=2nλi)2)1/2=(1+(1-λ1)2)1/2≤2.
It follows that ∥y∥0=1 and the claim is proved. Since {xn}n=1∞ has no convergent subsequences, one sees that co{xn}n≥1 is not relatively compact. This shows that B(X) is not nearly strictly convex.

3. The Representation and Continuity of Metric Projection onto a Hyperplane

Let x∈X and define
(11)σ(x)≔{=pC(x)2=pC∘(x*)2x*x*∈X*:x*(x)=pC(x)pC∘(x*)=pC(x)2=pC∘(x*)2},
which is analogous to the dual mapping in Banach spaces. Then, for x*∈X*, one obtains from the definition that σ-1(x*)={x∈X:x*∈σ(x)} and
(12)σ-1(λx*)=λσ-1(x*),∀λ∈R.
Hence, for 0≠x*∈X* and x∈X (noting that pC∘(x*)≠0), one has that
(13)x∈σ-1(x*)⟺x*(x)=pC(x)pC∘(x*)=pC∘(x*)2
and so
(14)x∈σ-1(x*pC∘(x*))⟺x*(x)pC∘(x*)=pC(x)=1.

Recall that the hyperplane H(x*,α) determined by x*∈X*∖{0} and α∈R is given by (5) and also that τC(x,H(x*,α)) is the distance from the point x to H(x*,α) defined by (3). The following result is an extension of the classical Ascoli Theorem for the distance formula under the case of norm from a pint to a hyperplane in a Banach space; see [3, Lemma 1.2, p. 24].

Proposition 5.

Let x*∈X*∖{0}, α∈R, and x∈X. Then
(15)τC(x,H(x*,α))=|α-x*(x)|pC∘(x*)sign(α-x*(x)).

Proof.

Without loss of generality, we assume that α>x*(x). Let y∈H(x*,α). Then x*(y)=α; hence
(16)α-x*(x)=x*(y-x)≤pC(y-x)pC∘(x*)
by Proposition 1(v). This implies that (α-x*(x))/pC∘(x*)≤pC(y-x) (noting that pC∘(x*)≠∅), and therefore
(17)α-x*(x)pC∘(x*)≤τC(x,H(x*,α))
because y∈H(x*,α) is arbitrary.

To show the converse inequality, let ϵ∈(0,pC∘(x*)). Then, by Proposition 1(iv), there is z∈C such that
(18)x*(z)>pC∘(x*)-ϵ.
Multiplying two sides of (18) by (α-x*(x))/x*(z)(pC∘(x*)-ϵ), one has that
(19)α-x*(x)pC∘(x*)-ϵ>α-x*(x)x*(z).
Now let y=x+((α-x*(x))/x*(z))z. Then, y∈H(x*,α), and
(20)pC(y-x)=α-x*(x)x*(z)pC(z)≤α-x*(x)x*(z)
because pC(z)≤1 by Proposition 1(ii) (noting that z∈C). It follows from (19) and (20) that (α-x*(x))/(pC∘(x*)-ϵ)>p(y-x). Hence, (α-x*(x))/(pC∘(x*)-ϵ)>τC(x,H(x*,α)). Letting ϵ→0 in this inequality gives
(21)α-x*(x)pC∘(x*)≥τC(x,H(x*,α)).
Thus the converse inequality of (17) follows. The proof is complete.

The first main result of this section is as follows, which gives a presentation of the generalized metric projection onto a closed hyperplane in X.

Theorem 6.

Let x*∈X*∖{0}, α∈R, and x∈X. Then the following assertion holds:
(22)PH(x*,α)C(x)={x+α-x*(x)pC∘(x*)2σ-1(x*),
if
σ-1(x*)≠∅,x*(x)<α,x+x*(x)-αpC∘(x*)2σ-1(-x*),
if
σ-1(-x*)≠∅,x*(x)>α.

Proof.

Similar to the proof of Proposition 5, we assume that σ-1(x*)≠∅ and x*(x)<α. Let y∈σ-1(x*). Then x*∈σ(y); hence
(23)x*(y)=pC(y)pC∘(x*)=pC(y)2=pC∘(x*)2.
Since pC∘(x*)≠0, one has that
(24)pC(y)=pC∘(x*).
Let z=x+((α-x*(x))/pC∘(x*)2)y. We then have from (23) that z∈H(x*,α) and from (24) and Proposition 5 that
(25)pC(z-x)=α-x*(x)pC∘(x*)2pC(y)=α-x*(x)pC∘(x*)=τC(x,H(x*,α)).
This implies that z∈PH(x*,α)C(x), and further
(26)PH(x*,α)C(x)⊇x+α-x*(x)pC∘(x*)2σ-1(x*).

To show the reverse inclusion, let z∈PH(x*,α)C(x). Then, z∈H(x*,α), and
(27)pC(z-x)=τC(x,H(x*,α))=α-x*(x)pC∘(x*)
thanks to Proposition 5. Noting that
(28)α-x*(x)pC∘(x*)=x*(z-x)pC∘(x*)≤pC(z-x),
we get from (27) that
(29)α-x*(x)=x*(z-x)=pC(z-x)pC∘(x*).
Now let y=(pC∘(x*)2/(α-x*(x)))(z-x). It follows from (29) that
(30)x*(y)=pC∘(x*)2α-x*(x)x*(z-x)=pC∘(x*)2,pC(y)=pC∘(x*)2α-x*(x)pC(z-x)=pC∘(x*).
Hence,
(31)x*(y)=pC(y)pC∘(x*)=pC∘(x*)2.
This means that y∈σ-1(x*) by (13), and so
(32)z=x+α-x*(x)pC∘(x*)2y∈x+α-x*(x)pC∘(x*)2σ-1(x*).
Consequently, PH(x*,α)C(x) is contained in the right-hand side of (26). The proof is complete.

The following result gives necessary and sufficient conditions for PH(x*,α)C(x)≠∅.

Proposition 7.

Let x*∈X*∖{0}, α∈R, and x∈X satisfy that x*(x)<α(resp.,x*(x)>α). Then PH(x*,α)C(x)≠∅ if and only if x*(resp.,-x*) attains its supremum pC∘(x*)(resp.,pC∘(-x*)) on bd C.

Proof.

Let x*,α, and x be as in Proposition 7, and let x*(x)<α. Suppose that PH(x*,α)C(x)≠∅. Take y∈PH(x*,α)C(x). Then x*(y)=α and pC(y-x)=τC(x,H(x*,α)). It follows from Propositions 5 and 1(v) that
(33)pC(y-x)=α-x*(x)pC∘(x*)=x*(y-x)pC∘(x*)≤pC(y-x).
Hence, x* attains its supremum pC∘(x*) at (y-x)/(pC(y-x))∈bd C.

Conversely, suppose that x* attains its supremum pC∘(x*) at x0∈bd C. Then x*(x0)=pC∘(x*) and pC(x0)=1; hence,
(34)x*(pC∘(x*)x0)=pC∘(x*)2=pC(pC∘(x*)x0)pC∘(x*).
This together with (13) implies that pC∘(x*)x0∈σ-1(x*), and therefore σ-1(x*)≠∅. By Theorem 6, one sees that PH(x*,α)C(x)≠∅. Similarly, we can prove another assertion for the case of x*(x)>α.

The second main result of this section is as follows, which describes the continuity of the generalized metric projection PH(x*,α)C onto the hyperplane H(x*,α) under the condition that the set C is weakly nearly strictly convex.

Theorem 8.

Let the set C be weakly nearly strictly convex, x*∈X*∖{0}, and α∈R. Then the following assertions hold.

Suppose that x∈X satisfies that x*(x)<α(resp.,x*(x)>α) and that x*(resp.,-x*) attains its supremum on bd C; then PH(x*,α)C is norm-to-weak upper semicontinuous at x.

If x* and -x* attain their supremum on bd C, then PH(x*,α)C is norm-to-weak upper semicontinuous on X. Furthermore, PH(x*,α)C is norm-to-norm upper semicontinuous at each point of H(x*,α).

Proof.

(i) Without loss of generality, we assume that x*∈X*∖{0}, x∈X, and α∈R satisfy x*(x)<α and that x* attains its supremum on bd C. We first show that σ-1(x*) is convex. To do this, let y1,y2∈σ-1(x*) and λ∈[0,1]. Then we obtain from (13) that
(35)x*(y1)=pC(y1)pC∘(x*)=pC∘(x*)2=pC(y2)pC∘(x*)=x*(y2).
This, together with Proposition 1(v) and (iii), implies that
(36)pC∘(x*)2=x*(λy1+(1-λ)y2)≤pC(λy1+(1-λ)y2)pC∘(x*)≤(λpC(y1)+(1-λ)pC(y2))pC∘(x*)=pC∘(x*)2.
Hence, λy1+(1-λ)y2∈σ-1(x*) by (13), and σ-1(x*) is convex.

We then show that σ-1(x*) is weakly compact. Since σ-1(x*/pC∘(x*))⊆bd C by (14) and since σ-1(x*/pC∘(x*))=(1/pC∘(x*))σ-1(x*) by (12), one sees that σ-1(x*/pC∘(x*)) is a convex subset of bd C. It follows that σ-1(x*/pC∘(x*)) is relatively weakly compact because C is weakly nearly strictly convex; hence, σ-1(x*) is relatively weakly compact. Thus, to complete the proof, it suffices to show that σ-1(x*) is weakly closed. To do this, let {xδ} be a net in σ-1(x*) convergent weakly to some x-∈X. Since
(37)x*(xδ)=pC(xδ)pC∘(x*)=pC∘(x*)2∀δ
and since pC is weakly lower semicontinuous by [15, Theorem 2.2.1, page 60], we have that
(38)pC∘(x*)2=x*(x-)=limδx*(xδ)=limδpC(xδ)pC∘(x*)≥pC(x-)pC∘(x*).
Noting that x*(x-)≤pC(x-)pC∘(x*), we get that
(39)x*(x-)=pC(x-)pC∘(x*)=pC∘(x*)2.
Hence, x-∈σ-1(x*) by (13), and the weak closedness of σ-1(x*) is proved.

Finally, we show that PH(x*,α)C is norm-to-weak upper semicontinuous at x. Otherwise, there exist a weakly open set
(40)W⊇PH(x*,α)C(x)
and a sequence {xn}⊆X with ∥xn-x∥→0 such that PH(x*,α)C(xn)⊈W. Since x*(x)<α and ∥xn-x∥→0, we may assume that each x*(xn)<α. Now take yn∈PH(x*,α)C(xn)∖W for each n. By Theorem 6, there exists zn∈σ-1(x*) such that yn=xn+((α-x*(xn))/pC∘(x*))zn for all n. Using the weak compactness of σ-1(x*), one has a subsequence {znk} of {zn} such that limkznk=z weakly for some z∈σ-1(x*). Therefore,
(41)
weak
-limkynk=y:=x+α-x*(x)pC∘(x*)z∈PH(x*,α)C(x).
This and (40) imply that y∈W. Since W is weakly open, one has that ynk∈W for sufficiently large k, which contradicts the choice of ynk and the proof of assertion (i) is complete.

(ii) Let x∈H(x*,α). Note that the norm-to-norm upper semicontinuity of PH(x*,α)C at x implies the norm-to-weak upper semicontinuity of PH(x*,α)C at x. It suffices to verify that PH(x*,α)C is norm-to-norm upper semicontinuous at x. To this end, let W be an open neighborhood of PH(x*,α)C(x)=x. Then there exists a positive number δ0 such that B(x,δ0)⊆W, where B(x,δ0) denotes the closed ball with center x and radius δ0. Below we show that there is δ∈(0,δ0] such that PH(x*,α)C(y)⊆B(x,δ0) whenever ∥y-x∥<δ or, equivalently (noting that if δ∈(0,δ0], one always has that PH(x*,α)C(y)=y∈B(x,δ0) whenever y∈H(x*,α) and y∈B(x,δ)),
(42)α-x*(x)pC∘(x*)2∥z∥≤δ0foreachz∈σ-1(x*)ifx*(y)<α,(43)x*(x)-αpC∘(x*)2∥z∥≤δ0foreachz∈σ-1(-x*)ifx*(y)>α
whenever ∥y-x∥<δ due to Theorem 6.

To proceed, we first verify
(44)∥z∥≤pC∘(x*)m1,∀z∈σ-1(x*),
where the positive number m1 is as in Proposition 1(vi). In fact, let z∈σ-1(x*). Then pC(z)=pC∘(x*) by (13). This, together with Proposition 1(vi), implies that ∥z∥≤(1/m1)pC(z)=(pC∘(x*)/m1), and (44) is proved. Next, take
(45)δ=min{m1δ0∥x*∥pC∘(x*),m1δ0∥x*∥pC∘(-x*),δ0}.
Then when ∥y-x∥<δ and x*(y)<α, one has, for each z∈σ-1(x*), that
(46)α-x*(x)pC∘(x*)2∥z∥=x*(x-y)pC∘(x*)2∥z∥≤∥x*∥∥x-y∥pC∘(x*)2∥z∥<∥x*∥δpC∘(x*)2pC∘(x*)m1≤δ0;
hence, (42) holds. While when ∥y-x∥<δ and x*(y)>α, we can similarly show that (43) is true. Thus, the proof of (ii) is complete.

A similar proof to that of Theorem 8 yields the following result.

Theorem 9.

Let the set C be nearly strictly convex, x*∈X*∖{0}, and α∈R. The the following assertions hold.

Suppose that x∈X satisfies that x*(x)<α (resp., x*(x)>α) and that x* (resp., -x*) attains its supremum on bd C; then PH(x*,α)C is norm-to-norm upper semicontinuous at x.

If x* and -x* attain their supremum on bd C, then PH(x*,α)C is norm-to-norm upper semicontinuous on X.

Theorem 10.

Suppose that the set C is strictly convex and that nonzero continuous linear functional x* and -x* attain, respectively, their supremum on bd C. Then PH(x*,α)C is norm-to-norm continuous on X.

Proof.

Let y*∈X*∖{0}. We assert that σ-1(y*) contains at most one point under the hypothesis made upon the set C. To do this, let y1,y2∈σ-1(y*). Then, from the proof of Theorem 8(i), one has that
(47)pC(λy1+(1-λ)y2)=pC(y1)=pC(y2)=pC∘(y*),∀λ∈[0,1];
hence, [y1/pC∘(y*),y2/pC∘(y*)]⊆
bd
C by Proposition 1(ii). It follows from the strict convexity of C that y1/pC∘(y*)=y2/pC∘(y*), that is, y1=y2, and that the assertion is proved. Applying this conclusion to x* and -x* (noting that σ-1(x*) and σ-1(-x*) are nonempty by (13) because x* and -x* attain their supremum on bdC), one sees that both σ-1(x*) and σ-1(-x*) are a singleton. Therefore, PH(x*,α)C(x) is single valued for each x∈X by Theorem 6, and PH(x*,α)C is norm-to-norm continuous on X by Theorem 9, which completes the proof of Theorem 9.

Theorem 11.

Suppose that X is reflexive and that the set C is strictly convex. Then PH(x*,α)C is norm-to-norm continuous on X for each x*∈X*∖{0}.

Proof.

Let x*∈X*∖{0} be arbitrary. Below we will show that x* attains its supremum on bdC. Granting this, the conclusion follows from Theorem 10. To this end, we take a sequence {xn}⊆C such that limnx*(xn)=pC∘(x*). Since the set C is weakly compact (noting that X is reflexive), there exists a subsequence of {xn}, denoted still by {xn}, such that limnxn=x weakly for some x∈C. Thus,
(48)pC∘(x*)=x*(x)≤pC(x)pC∘(x*)≤pC∘(x*).
Consequently, x∈bdC because pC(x)=1, and x* attains its supremum at x. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author was supported in part by the NNSF of China (Grant no. 11271342) and the NSF of Zhejiang Province (Grant no. LY12A01029).

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