ON LOWER SEMICONTINUITY OF THE RESTRICTED CENTER

. Given a ﬁnite dimensional subspace V and a certain family F of nonempty closed and bounded subsets of C 0 ( T, U ), where T is a locally compact Hausdorﬀ space and U is a strictly convex Banach space, we investigate here lower semicontinuity of the restricted center multifunction C V : F →→ V. In particular, we establish a Haar-like intrinsic characterization of ﬁnite dimensional subspaces V of C 0 ( T, U ) which yields lower semicontinuity of C V .


INTRODUCTION
Let us be given a family F of nonempty closed and bounded subsets of a normed linear space X, and a finite dimensional subspace V of X.For F ∈ F and x ∈ X, let r(F ; x) := sup{ x − y : y ∈ F } denote the radius of the smallest closed ball centered at x covering F and let The number r V (F ) is called the restricted (Chebyshev) radius of F in V.It is easily seen that the set C V (F ) is nonempty, closed and convex.A typical element v 0 ∈ C V (F ) is usually called a restricted (Chebyshev) center or a best simultaneous approximant of F in V.The multifunction C V : F → → V, with values C V (F ), F ∈ F, is called the restricted center multifunction.
Let us note that in case F is a singleton {x}, r V (F ) is the distance of x from V, denoted by d(x, V ), and C V (F ) is the set of best approximants to x in V.The multifunction P V : X → → V, in this case, is usually called the metric projection onto V .The problems concerning various continuities of metric projection in the Banach space C 0 (T ) of real-valued continuous functions vanishing at infinity have been significantly investigated by a number of authors (cf., e.g., [2], [3], [5], [9], [6], [7], [12], [16]).Some of these results pertaining to lower semicontinuity of metric projection have been generalized to the space C 0 (T, U ), where U is a strictly convex Banach space in ( [4], [11], [12]).
Given a finite dimensional subspace V of an arbitrary real normed linear space X, we investigate here a sufficient condition for lower semicontinuity of the restricted center multifunction C V defined on a certain family F of subsets of X.This extends certain results in [4] for lower semicontinuity of metric projection.In [8], a characterization of lower semicontinuity of restricted center multifunction defined on a certain family F of subsets of the space C 0 (T ) (Theorem 3.2) was studied.This result was in the same spirit as that of the particular case of ( [11], Theorem 4.1) for metric projections.Here we extend this investigation to the space C 0 (T, U ).This approach naturally leads us to our main goal of exploring an intrinsic Haar-like characterization of finite-dimensional subspaces V of C 0 (T, U ) , for which the restricted center multifunction

PRELIMINARIES
Throughout the following, X will be a real normed linear space which for the most part will be the Banach space C 0 (T, U ), where T is a locally compact Hausdorff space and U is a strictly convex (real) Banach space, and V will be a finite dimensional subspace of X.
Let us recall that C 0 (T, U ) consists of all continuous functions f : T → U vanishing at infinity, i.e, a continuous function f is in C 0 (T, U ) if and only if, for every > 0, the set {t ∈ T : f (t) ≥ } is compact.The space C 0 (T, U ) is endowed with the norm: Throughout the remainder, V will be a finite dimensional subspace of X.
Let CLB(X) denote the family of all nonempty closed and bounded subsets of X equipped with the Hausdorff metric H defined by where e(A, B) := sup{d(a, B) : a ∈ A} denotes the excess of A over B.
If F ⊆ CLB(X), we regard F as a metric space equipped with the induced Hausdorff metric topology.By a multifunction T : F → → V we mean a setvalued function whose values T (F ), F ∈ F are nonempty closed subsets of V .Recall that a multifunction T : F → → V is said to be lower semicontinuous (resp.upper semicontinuous) abbreviated lsc (resp.usc) if the set {F ∈ F : It follows immediately from the definitions that T is lsc if and only if T = T * .Next, let us recall [17] that a set F ∈ CLB(X) is said to be sup-compact w.r.t.V if for each v 0 ∈ V, every maximizing sequence {f n }, i.e., a sequence [21].Examples of sets which are sup-compact but not compact are also given there.Let In the sequel, for some of the results to follow, we will take F = s-K V (X) which contains the family K V (X) of all nonempty compact subsets F of X satisfying the same restriction r V (F ) > r X (F ).

A SUFFICIENT CONDITION FOR LOWER SEMICONTINUITY OF THE
MULTIFUNCTION C V Throughout this section X will be a (real) normed linear space whose normed dual will be denoted by X * , and V will be a finite dimensional subspace of X.The weak * or σ(X * , X)-topology of X * will be denoted by w * .Let Ext(B(X * )) denote the set of all extreme points of the closed unit ball B(X * ) of X * .For the sake of brevity, let us denote the closure being taken in the w * -topology.Also, for x ∈ X, let denote the set of all critical functionals.Clearly, E x is nonempty and w *compact subset of X * for each x ∈ X.For A ⊆ X, we denote by A ⊥ the annihilator of A : For f ∈ X, let For F in CLB(X), let us denote by G F , the subspace Note that We will denote the relative interior of C V (F ) by relintC V (F ).
Lemma 1.Let V be a finite dimensional subspace of a normed space X, and let F ∈ CLB(X) be sup-compact w.r.t.V.
Proof.Let v 0 ∈ relintC V (F ).Then there exists > 0 such that for every , and we conclude that Theorem 2. Let X, V and F be as in Lemma 1.
such that H(G, F ) < min{ , δ} and any g ∈ G, we have ( 4) sup From ( 4), ( 5) and ( 6), it follows that r(G, where T is a locally compact Haussdorff space, and U is a strictly convex (real) Banach space.Throughout the remainder, V will be a finite dimensional subspace of X.
denote the set of all critical points of the function f − α.Also let We note that in case X = C 0 (T, U ) the set of extreme points of the closed unit ball of X * is given by (cf., e.g., [15], p.422), ExtB(X * ) = {x * u * ,t : u * ∈ ExtB(U * ), t ∈ T }, where x * u * ,t (x) = u * (x(t)), x ∈ X.Also note that in this case if U * is also assumed to be strictly convex, then in the above representation of ExtB(X * ), we may take u * in S(U * ), the unit sphere of U * .
The following theorem for characterization of restricted centers whose proof follows easily from ([17], Theorem 2.6) and the above representation of the extreme points of B(X * ) will be required as a tool in the sequel.
The following statements are equivalent.
(iii) The origin (0, . . ., 0) of R n belongs to the convex hull co(S) of S, where The next lemma is an analogue of Lemma 1 for the present case.Its proof is exactly identical.Let us recall that we are denoting by G F the subspace and by relintC V (F ), the relative interior of C V (F ).
Lemma 5. Let X, V and F be as in the last theorem.
Hence, the conclusion of the lemma can be restated as follows: If 0 ∈ relintC V (F ), then for every f 0 ∈ Q F,0 .
4.1.An intrinsic characterization of lower semicontinuity of the multifunction C V .As before, let X = C 0 (T, U ) and V be a finite dimensional subspace of X.The next lemma involves perturbation of sets.For F, G in K(X), and S ⊆ T, we write F | S = G| S if for every f ∈ F, there is a g ∈ G such that f | S = g| S , and conversely.The proof is a verbatim reproduction of Lemma 2 in [8] which was given for C 0 (T ).However, for the convenience of the reader, we give it once again here.
Let us now recall the following well known result for lower semicontinuity of metric projection due to Blatter, Morris and Wulbert [2].
Theorem 8. Let X = C(T ) and V be a finite dimensional subspace of X.

The metric projection multifunction P
We are now ready to state and prove our first main characterization theorem for lower semicontinuity of C V .This extends Theorem 2 of [8] and Theorems 6 and 9 of [4].
Proof.(i) For every F ∈ K(X) with 0 ∈ relintC V (F ), C V is given to be lsc at F. If possible let (i) be not true, i.e., suppose there exists G ∈ K(X), v 0 ∈ relintC V (G) and an element By Lemma 5, E(f 0 ) ⊆ Z(C V (F )).Hence there exists a t 0 ∈ E(f 0 ) such that t 0 ∈ bdZ(C V (F )), i.e., there exists a net {t λ : λ ∈ Λ} such that t λ ∈ Z(C V (F )), λ ∈ Λ, and Two cases arise as follows.Case (i) Since V is finite-dimensional, if necessary by passing to a subnet, we may assume that for each λ ∈ Λ there exists some If necessary by passing once more to a subnet, it can be ensured that there are signs k ∈ {−1, 1}, k = 1, . . ., m, such that we have . ., m.For each δ > 0, the set B δ = {t ∈ T : |f 0 (t 0 ) − f 0 (t)| < δ} is a neighborhood of t 0 .Hence there exists λ ∈ Λ such that t λ ∈ B δ .Since t λ ∈ Z(C V (F )), t λ ∈ E(f 0 ).Since Z(C V (F )) is a closed set, there exists a compact neighborhood W of t λ such that Z(C V (F )) ∩ W = ∅.Without loss of generality, we may assume that W ⊂ B δ .Let ρ be a continuous function such that 0 ≤ ρ(t) ≤ 1 for t ∈ T, ρ(t λ ) = 1 and ρ(t) = 0 for t ∈ T \ W. Define We now have an

We now define an open set
In the above set of inequalities, t λ is a fixed element of the net If the net {t λ : λ ∈ Λ} does not satisfy the conditions that we assumed in the first case, then we need to consider the following alternative.Case (ii) By passing to a subnet, if necessary, we may assume that u * (v 1 (t λ )) = 0 for all λ ∈ Λ and u * ∈ E f 0 (t 0 ) .
If necessary by passing once more to a subnet, it can be ensured that there are signs k ∈ {−1, 1}, k = 1, . . ., m, such that we have For δ > 0, the set is a neighborhood of t 0 .Hence there exists λ ∈ Λ such that t λ ∈ B δ .Also, there exists a compact neighborhood W of t λ such that W ∩Z(C V (F )) = ∅.We may assume, without loss of generality, that W ⊂ B δ .Let ρ be a continuous function such that 0 ≤ ρ(t) ≤ 1 for t ∈ T, ρ(t λ ) = 1 and ρ(t) = 0 for t ∈ T \ W.
Let us define Then f δ ∈ C 0 (T, U ), and using (11), for t ∈ B δ , we have it is easy to see that f δ = f 0 .Thus again defining F δ := F ∪ {f δ }, we have F δ ∈ K(X) for each δ > 0 and H(F, F δ ) < δ.Exactly as in case(i) we again have an F ∈ K(X) with 0 ∈ relintC V (F ) and

Consider again the open set
Therefore, C V (F δ ) ∩ A = ∅.Thus in either case the hypothesis that the multifunction C V is lsc at F is contradicted, and we conclude that contrary to our assumption, E(g 0 − v 0 ) ⊆ int Z(G G ) must hold.This completes the proof of (i).
(ii) The proof of Theorem 2 (ii) of [8] extends verbatim to the present case.
We can now state a global necessary and sufficient condition for lower semicontinuity of C V as follows.
Theorem 10.Let V be a finite dimensional subspace of X = C 0 (T, U ). Then the multifunction C The next theorem partially extends Theorem 4.5 of Blatter, Morris, and Wulbert [2].
Theorem 11.Let X, V as in the last theorem.If for every F ∈ K(X) with 0 ∈ relintC V (F ), the set both open as well as closed, the function g 0 : T → U defined by Hence, , and by Theorem 2, we conclude that C V is lsc at F.

4.2.
Characterization of lower semicontinuity of C V using Haar-like condition.Our goal here is to give an intrinsic characterization of finite dimensional subspaces V of C 0 (T, U ) for which the restricted center multifunction C V : K(X) → → V is lsc.Let us recall that a finite dimensional subspace V of C 0 (T ) is called a Haar subspace (or that it is said to satisfy Haar condition) if for each v ∈ V \ {0}, cardZ(v) ≤ dim V −1.It is easily seen that V is a Haar subspace of dimension n if and only if dim V | S = n for every subset S of T such that card(S) = n.
Here card(S) denotes the cardinality of S and V | S := {v| S : v ∈ V }.Let us also recall the generalized Haar condition introduced by Zukhovitskii and Stechkin [22] for a finite dimensional subspace V of C 0 (T, U ). Consider the following properties of V .
(T m ) For each v ∈ V \ {0}, there are at most m zeros in T.
(P m ) For each set of m distinct points t i ∈ T and m elements u i ∈ U, there exists at least one v ∈ V , such that An n-dimensional subspace V of C 0 (T, U ) is said to satisfy the generalized Haar condition if either dim U = k ≤ n and V satisfies conditions (T m ) and (P m ) where m ∈ N is the unique integer satisfying mk < n ≤ (m + 1)k, or dim U > n, and V satisfies condition (T 0 ).It is easily seen that in case dim U = k < ∞, V satisfies conditions (T m ) and (P m ) if and only if for any finite set S ⊆ T, dim V | S ≥ min{dim V, k.card(S)}.For finite dimensional subspaces V of C 0 (T ), the following extension of Haar condition is due to W. Li [10].
For finite dimensional subspaces V of C 0 (T, U ), the following variant of the generalized Haar condition is also due to W. Li [12].
Note that if T is connected, then property (Li) coincides with the Haar condition.Moreover, in case dim U = k ≤ n = dim V , the property (Li ) is implied by the generalized Haar condition.
We require the following lemma due to W. Li [12] to prove the next theorem.
Lemma 14.For a finite dimensional subspace V of C 0 (T, U ), the following statements are equivalent.
(i) The metric projection multifunction P The next theorem gives an intrinsic characterization of finite dimensional subspaces V of C 0 (T, U ) for which the restricted center multifunction C V is lsc.It generalizes Theorem 3 of [8].
Theorem 15.For a finite dimensional subspace V of C 0 (T, U ), the following statements are equivalent.
Proof.We imitate here the proof of Thorem 3 of [8].(i) ⇒ (ii) : The statement (i) restricted to singletons is nothing but the lower semicontinuity of metric projection P V .It follows from Lemma 14 that the lower semicontinuity of metric projection P V is equivalent to V satisfying property (Li ).Hence (ii) is true.(ii) ⇒ (i): In view of Theorem 9 (ii), without loss of generality, it is enough to prove that property (Li ) gives, ( 12) for all f 0 ∈ Q F,0 , whenever F ∈ K V (X) and 0 ∈ relintC V (F ).We prove this by the method of contradiction.Assume (12) does not hold for some F ∈ K V (X) and some f 0 ∈ Q F,0 , where 0 ∈ relintC V (F ).For simplicity, we denote int Z(C V (F )) by M .Therefore, we have and we get equality in ( 13).Hence we have, = r V 0 (F 0 ).
From the above, it also follows that A f 0 = {t ∈ T 0 : f 0 (t) = f 0 T 0 }, i.e., A f 0 = E(f 0 ) ∩ T 0 .Hence A f 0 is none other than the set of all critical points of 0 with f 0 | T 0 as the farthest point.
Since V is finite dimensional, this in turn implies that, {t 1 , . . ., t m } ⊆ int This contradicts the definition of T 0 .Hence E(f 0 ) ⊆ intZ(C V (F )) must hold for every f 0 ∈ Q F,0 , 0 ∈ relintC V (F ) and F ∈ K V (X).We can now apply Theorem 9 (ii) to conclude that the multifunction C V is lsc.
As observed in Theorem D and Theorem 3 of Li [12], in case U is a strictly convex Banach space such that dim U = k ≤ n, and V is an n-dimensional subspace of X, the generalized Haar condition consisting of (P m ) and (T m ) where m ∈ N is the unique integer satisfying mk < n ≤ (m + 1)k is equivalent to the condition (16) cardZ(v)) ≤ (dim U ) −1 • dim{p| bdZ(v) : p ∈ V and p| int Z(v) = 0}.
In conjunction with Theorem 3.5 of [17], this result yields the next theorem.
Theorem 16.Let U be a k-dimensional Euclidean space and V be an ndimensional subspace of X = C 0 (T, U ).If k ≤ n, then in order that C V (F ) be a singleton for each F ∈ K(X) it is necessary and sufficient that condition (16) be satisfied.