Proximinality in Banach spaces

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Proximinality in Banach spaces

✩

Pradipta Bandyopadhyay

_{, Yongjin Li}

_{, Bor-Luh Lin}

c,∗

_{, Darapaneni Narayana}

d a_{Stat-Math Division, Indian Statistical Institute, 203, B.T. Road, Kolkata 700 108, India}

b_{Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, PR China} c_{Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA} d_{Department of Mathematics, Indian Institute of Science, Bangalore 560012, India}

Received 11 April 2007 Available online 18 October 2007

Submitted by J. Bastero

Abstract

In this paper, we study approximativelyτ-compact andτ-strongly Chebyshev sets, whereτis the norm or the weak topology. We show that the metric projection ontoτ-strongly Chebyshev sets are norm-τ continuous. We characterize approximatively

τ-compact andτ-strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces inτ-almost locally uniformly rotund spaces. We also prove some stability results on approximativelyτ-compact andτ-strongly Chebyshev subspaces.

Keywords:Approximativelyτ-compact;τ-Strongly Chebyshev; Metric projection;τ-Almost locally uniformly rotund

1. Introduction

LetX be a real Banach space. For a closed setK inX andx∈X, we denote the distance function of K atx

by d(x, K)=inf{x−k: k∈K}. The metric projection of x ontoK isPK(x)= {k∈K: x−k =d(x, K)}.

The setK is called proximinal (respectively Chebyshev) if for every x∈X\K,PK(x)is nonempty (respectively

a singleton). It is known that a Banach spaceXis reflexive if and only if every closed convex subset ofXis proximinal inX.

The notion of approximative compactness has been introduced by Efimov and Stechkin [6] (see also [4]). Deutsch [5] extended this notion to define approximatively τ-compact sets for a “regular mode of convergenceτ” [5, Definition 2.3] whereτ includes the norm, weak or weak∗topology. However, there are many cases in whichτ

does not arise from any topology.

✩ _{The work of the first author was partially supported by IFCPAR grant no. 2601-1 & DST-NSF grant no. RPO-141. The work of the second} author was partially supported by the Foundation of Sun Yat-Sen University Advanced Research Centre and Lingnan Foundation. The work of the third author was partially supported by NSF grant no. NSF-OISE-03-52523.

* _{Corresponding author.}

E-mail addresses:[email protected] (P. Bandyopadhyay), [email protected] (Y. Li), [email protected] (B.-L. Lin), [email protected] (D. Narayana).

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In this paper, unless otherwise mentioned, we denote byτ either the norm or the weak topology onX. And we consider the following related notions of proximinality. Recall that a sequence{zn} ⊆K is called minimizing for

x∈X\K, ifx−zn →d(x, K).

Definition 1.1.LetKbe aτ-closed subset ofXandx0∈X\K.

(a) We say that K is approximativelyτ-compact forx0 if every minimizing sequence {zn} ⊆K for x0 has a τ

-convergent subsequence.

(b) We say thatKisτ-strongly Chebyshev forx0if every minimizing sequence{zn} ⊆Kforx0isτ-convergent.

IfKis approximativelyτ-compact (orτ-strongly Chebyshev) for everyx∈X\K, we sayKis approximatively

τ-compact (respectivelyτ-strongly Chebyshev) inX. As usual, in caseτ is the norm topology, we omit it.

We note that approximativeτ-compactness implies proximinality. It is also clear that ifKisτ-strongly Chebyshev forx0, thenPK(x0)is a singleton.

Any compact set or any finite-dimensional subspace of a Banach space is clearly approximatively compact. Simi-larly, any weakly compact set or any reflexive subspace of a Banach space is clearly approximatively weakly compact.

The notion of strongly proximinal subspaces have been defined in [8]. We extend this to define the following:

Definition 1.2.For a closed setK⊆X,x∈Xandδ >0, let

PK(x, δ)=z∈K:z−x< d(x, K)+δ.

A τ-closed set K ⊆X is τ-strongly proximinal for x ∈X \K, if K is proximinal for x and for any τ -neighbourhoodV of 0 inX, there existsδ >0 such thatPK(x, δ)⊆PK(x)+V. If this holds for everyx∈X\K,

thenKis said to beτ-strongly proximinal inX.

Clearly, whenKis a subspace andτ is the norm topology, we get back the definition of [8].

In Section 2, we study the relationship between approximativelyτ-compact,τ-strongly proximinal andτ-strongly Chebyshev sets and prove that the metric projection ontoτ-strongly Chebyshev sets are norm-τ continuous. We also characterize approximativelyτ-compact andτ-strongly Chebyshev hyperplanes as kernels ofτ-strongly support functionals (see Definition 2.9) andτ-strongly exposing functionals, respectively. We show that ifY is a reflexive Chebyshev subspace ofXsuch thatP_Y−1(0)is weakly closed, thenPY is weak–weak continuous.

In Section 3, we consider proximinality inτ-almost locally uniformly rotund (τ-ALUR) Banach spaces. These spaces were defined in [2] (see Definition 3.1). From the results of [1], it follows that a Banach space X is τ -ALUR⇔every norm-attaining functionalx∗∈X∗is aτ-strongly exposing functional. We refer to [1,2] for various characterizations and properties of such spaces.

It is well known that if a finite-codimensional subspaceY ofX is proximinal inX, thenY⊥⊆NA(X), the set of all norm-attaining functionals inX∗. Conditions under which the converse holds have been the subject of many recent papers, e.g., [8,9,14]. We show that the converse holds inτ-ALUR spaces and, in fact, implies thatY isτ-strongly Chebyshev.

In Section 4, we consider some stability results on approximativelyτ-compact andτ-strongly Chebyshev sub-spaces.

The closed unit ball and the unit sphere ofXwill be denoted byBXandSX, respectively. We will denote byNA(X)

the norm-attaining functionals inX∗. For a closed bounded convex setC, extCdenotes the set of extreme points ofC. By a subspace, we mean a closed linear subspace.

2. General results

Note that ifKisτ-closed, thenKisτ-strongly Chebyshev⇒Kis approximativelyτ-compact⇒Kis proximinal. And none of the implications can be reversed (see Example 2.1 and Theorem 2.8).

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Indeed, if a minimizing sequence{zn}forx has aτ-convergent subsequence{znk}withτ-limit pointz∈K, then,

since the norm isτ-lower semicontinuous,

d(x, K)x−zlim infx−znk =d(x, K).

Thus,z∈PK(x).

Example 2.1.LetX=ℓ∞,Y =c0. ThenY, being anM-ideal inX, is proximinal inX. Forx0=(1,1,1, . . .)∈ℓ∞,

the sequenceyn=(1,1, . . . ,1,0,0, . . .)∈Y is minimizing, but{yn}has noτ-convergent subsequence. Thus

proxim-inality does not imply approximativeτ-compactness.

As suggested by the name,τ-strongly Chebyshev sets are precisely sets that areτ-strongly proximinal and Cheby-shev. Indeed, we have

Theorem 2.2.LetKbe aτ-closed subset of Banach spaceXandx0∈X\K. ThenKis approximativelyτ-compact

forx0⇔Kisτ-strongly proximinal forx0andPK(x0)isτ-compact.

Proof. Notice that inτ-topology, compactness and sequential compactness coincide. It follows that ifKis approxi-mativelyτ-compact forx0, thenPK(x0)isτ-compact.

IfK is notτ-strongly proximinal forx₀, then there exist a τ-neighbourhoodV of 0 and a minimizing sequence

{zn} ⊆K withzn∈/PK(x0)+V for alln1. SinceK is approximativelyτ-compact forx0, there is a subsequence {znk}such thatznk→z0∈Kinτ. Thenz0∈PK(x0)and so,zn∈z0+V ⊆PK(x0)+V for somen1. A

contra-diction!

Conversely, supposeK isτ-strongly proximinal forx0andPK(x0)isτ-compact, butK is not approximatively τ-compact forx0. Then there is a minimizing sequence{zn} ⊆Ksuch that no subsequence isτ-convergent. It follows

that for any z∈PK(x0), there is a τ-neighbourhoodUz of zandNz∈Nsuch that for all nNz,zn∈/Uz. Since

PK(x0)is τ-compact, there is aτ-neighbourhoodV of 0 andN0∈Nsuch that for allnN0,zn∈/PK(x0)+V.

Since K isτ-strongly proximinal for x0, there existsδ >0 such thatPK(x0, δ)⊆PK(x0)+V. Note that for any

minimizing sequence{zn} ⊆K,zn∈PK(x, δ)eventually. This is a contradiction! ✷

Theorem 2.3.LetKbe aτ-closed subset of Banach spaceXandx0∈X\K. Then the following are equivalent:

(a) Kisτ-strongly Chebyshev forx0.

(b) Kisτ-strongly proximinal forx0andPK(x0)is a singleton.

Proof. By Theorem 2.2, it suffices to show(b)⇒(a).

SupposeK isτ-strongly proximinal forx0 andPK(x0)= {z0}. Let V be a τ-neighbourhood of 0. SinceK is τ-strongly proximinal forx0, there existsδ >0 such thatPK(x0, δ)⊆z0+V. Thus, for any minimizing sequence {zn} ⊆K,zn∈PK(x, δ)⊆z0+V for sufficiently largen. Hencezn→z0inτ. ✷

Example 2.4.It is shown in [11, Proposition IV.1.14] that if Y is a proximinal subspace of X such that for every

x∈X,PY(x)is weakly compact, then

card(extBX/Y)card(extBX).

ForX=c0orL1(μ)withμnonatomic, since extBX= ∅, it follows thatXhas no Chebyshev or approximatively

τ-compact subspace of finite-codimension. Example 2.1 is a special case of this general phenomenon. On the other hand, it is known that any proximinal subspace of finite-codimension inc0is strongly proximinal [8, Theorem 3.4].

Thus, strong proximinality does not imply approximativeτ-compactness.

And, for X =C_R([0,1]), since card(extBX)=2, it follows that X has no Chebyshev or approximatively τ

-compact subspace of finite-codimension2 (also see [15, pp. 321–324]). Regarding continuity of the metric projection, we have the following result.

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Theorem 2.5.LetKbe aτ-closed subset of a Banach spaceXandx0∈X\K.

(a) Kisτ-strongly Chebyshev forx0⇒the metric projectionPK is single-valued and norm-τ continuous atx0, i.e.,

ifxn→x0in norm andzn∈PK(xn), thenzn→z0=PK(x0)inτ.

(b) Kis approximativelyτ-compact forx0⇒the metric projectionPKis norm-τ upper semicontinuous(usc)atx0,

i.e., for anyτ-open setW withPK(x0)⊆W, there existsε >0such thatPK(x)⊆W wheneverx−x0< ε.

Proof. (a)IfK isτ-strongly Chebyshev forx0, thenPK(x0)is a singleton, say{z0}. Observe that, ifxn→x0and zn∈PK(xn), then{zn}is a minimizing sequence forx0. Therefore,zn→z0inτ-topology.

(b)AssumeKis approximativelyτ-compact forx₀. Then by Theorem 2.2,Kisτ-strongly proximinal forx₀and

PK(x0)isτ-compact.

LetW be aτ-open set such thatPK(x0)⊆W. SincePK(x0)isτ-compact, there exists aτ-open neighbourhoodV

of 0 such thatPK(x0)+V ⊆W. SinceK isτ-strongly proximinal forx0, there existsδ >0 such thatPK(x0, δ)⊆ PK(x0)+V ⊆W. Clearly, forx−x0< δ/2,PK(x)⊆PK(x0, δ)⊆W. ✷

Remark 2.6.Letτ be a topology onXsuch thatBXisτ-closed. This corresponds to a “regular mode of convergence”

that is “topological” [5]. Examples include, apart from the norm, weak and weak∗ _{topologies, the strong and weak}

operator topologies on the spaceL(X, Y ). If we define approximativelyτ-compact sets and τ-strongly Chebyshev sets in terms of minimizing nets rather than sequences, the modified definition forτ-strongly Chebyshev sets is easily seen to be equivalent to the original one. And all the above results still hold. If a “regular mode of convergence” is not “topological,” we do not know if Theorems 2.2 and 2.5(b)will still hold.

compact. We do not know if the converse hold in general. Still, aτ-closed setK is approximativelyτ-sequentially compact forx₀∗⇒the metric projectionPK is norm-τ usc forx₀∗. This follows from [5, Theorem 2.7] and gives an

alternative proof of Theorem 2.5(b). Whenτ is the norm topology, we have

Proposition 2.7.LetXbe a Banach space,Kbe a closed subset ofXandx₀∈X\K. ThenKis strongly Chebyshev forx₀⇔for everyε >0, there existsδ >0such that · -diam(PK(x0, δ)) < ε.

Proof. IfKis strongly Chebyshev forx0, thenPK(x0)= {z0}. Therefore every minimizing sequence converges toz0.

Now, if the given condition does not hold, there existsε >0 andzn∈PK(x0,1/n)such thatzn−z0ε. Then{zn}

is a minimizing sequence that does not converge toz0. Contradiction!

Conversely, if for everyε >0, there existsδ >0 such that · -diam(PK(x0, δ)) < ε, then the Cantor Intersection

Theorem shows thatPK(x0)is a singleton{z0}(say). Suppose{zn} ⊆Kis a minimizing sequence forx0. Letε >0

and letδ >0 be such that · -diam(PK(x0, δ)) < ε. Then, for all sufficiently large n,zn∈PK(x0, δ). Therefore, zn−z0 · -diam(PK(x0, δ)) < ε. ✷

norm continuous for all nonempty closed convex setsK⊆Xand they have given various characterizations for such spaces. The following result characterizes reflexive spaces from proximinality point of view. The proof follows from arguments in [5,7].

Theorem 2.8.LetXbe a Banach space.

(a) Every closed convex set is proximinal inX⇔Xis reflexive⇔every closed convex set is approximatively weakly compact inX.

(b) Every closed convex set is weakly strongly Chebyshev inX⇔Xis reflexive and strictly convex. (c) Every closed convex set is approximatively compact inX⇔Xis reflexive and has the KK property.

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A.L. Brown [3] has shown that there exists a separable strictly convex and reflexive real Banach spaceX and a closed subspaceY such thatPY is not norm–norm continuous.

We now characterize approximativelyτ-compact andτ-strongly Chebyshev hyperplanes.

Definition 2.9.

(a) Recall that x∗ ∈SX∗ is called a τ-strongly exposing functional if x∗∈NA(X) and every {x_n} ⊆B_X with

limx∗(xn)=1 isτ-convergent.

(b) Let us callx∗∈SX∗aτ-strongly support functional ifx∗∈NA(X)and every{x_n} ⊆B_Xwith limx∗(x_n)=1 has

aτ-convergent subsequence.

Theorem 2.10.For a Banach spaceX andx∗∈SX∗,kerx∗ is approximativelyτ-compact(respectivelyτ-strongly

Chebyshev)inX⇔x∗is aτ-strongly support(respectivelyτ-strongly exposing)functional.

Proof. SupposeY=kerx∗is approximativelyτ-compact. ThenY is proximinal, and hence,x∗∈NA(X). Thus, there existsx0∈SXsuch thatx∗(x0)=1. Let{xn} ⊆SXsuch thatx∗(xn)→1. Letyn=x∗(xn)x0−xn∈Y. Then

1=x∗(x0)=x∗(x0−yn)x0−yn =x₀−x∗(xn)x0+xn

1−x∗(xn)+1→1.

Thus,{yn}is a minimizing sequence forx0, and hence has aτ-convergent subsequence. It follows that so does{xn}.

Conversely, suppose x∗ is a τ-strongly support functional. Then, x∗ ∈NA(X), and hence, Y =kerx∗ is a proximinal subspace. Indeed, let x0∈SX be such that x∗(x0)=1. Then for any x ∈X, d(x, Y )= |x∗(x)| and x−x∗(x)x0∈PY(x).

Let {zn} ⊆Y be a minimizing sequence for x∈X\Y. It follows that x−zn →d(x, Y )= |x∗(x)| =0 and

x∗(x−zn)=x∗(x). Letwn=ηx(x−zn)/x−zn, whereηx is the sign of x∗(x), i.e.,ηxx∗(x)= |x∗(x)|. Then

wn =1 and the sequence x∗(wn)→1. Sincex∗is aτ-strongly support functional, the sequence{wn}has aτ

-convergent subsequence. And hence,{zn}has aτ-convergent subsequence too.

The other case is similar. ✷

The following is the analogue of Theorem 2.8 without reflexivity.

Theorem 2.11.For a Banach spaceX, the following are equivalent:

(a) Everyx∗∈NA(X)is aτ-strongly support(respectivelyτ-strongly exposing)functional.

(b) Every proximinal convex setK⊆Xis approximativelyτ-compact(respectivelyτ-strongly Chebyshev). (c) Every proximinal subspaceY⊆Xis approximativelyτ-compact(respectivelyτ-strongly Chebyshev). (d) Every proximinal hyperplane is approximativelyτ-compact(respectivelyτ-strongly Chebyshev).

Proof. (a)⇒(b). Suppose everyx∗∈NA(X)is aτ-strongly support functional. LetK be a proximinal convex set inX andx0∈X\K. Without loss of generality, we may assumex0=0 andd(0, K)=1. Then there existsz0∈K

such thatz₀ =1=d(0, K). Sinced(0, K)=1,Kand the open unit ball ofXare disjoint convex sets and therefore, there existsx∗∈SX∗such that inf_x∗_(K)1. It follows thatx∗(z0)=1, that is,x∗∈NA(X). Thus,x∗is aτ-strongly

support functional. Let{zn} ⊆Kbe a minimizing sequence for 0. Then 1x∗(zn)zn →1. Therefore,{zn/zn},

and hence{zn}, has aτ-convergent subsequence.

(b)⇒(c)⇒(d)is trivial, and(d)⇒(a)follows from Theorem 2.10. ✷

Remark 2.12.Observe that a subspaceY need not be strongly Chebyshev even ifY is a Chebyshev subspace ofX

andPY is norm–norm continuous. For example, letx∗∈SX∗ be an exposing functional that is not strongly exposing.

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Definition 2.13.Forx∗∈X∗\ {0}andx∈X, let us define the following maps:

D(x)=x∗∈SX∗: x∗(x)= x, D−1x∗=x∈SX: x∗(x)=x∗

Dis called the duality map andD−1is called the pre-duality map. Naturally,D−1is defined only onNA(X).

Theorem 2.14.LetXbe a Banach space andx₀∗∈SX∗.

(a) x₀∗is aτ-strongly exposing functional⇒the pre-duality mapD−1is single-valued and norm-τ continuous atx₀∗, i.e., ifx_n∗→x₀∗in norm andxn∈D−1(xn∗), thenxn→x0=D−1(x0∗)inτ.

(b) x₀∗is aτ-strongly support functional⇒the pre-duality mapD−1is norm-τ usc atx₀∗, i.e., for anyτ-open setW

withD−1(x₀∗)⊆W, there existsε >0such thatD−1(x∗)⊆W wheneverx∗−x₀∗< ε.

Proof. Observe that ifx₀∗∈NA(X)andY =kerx₀∗, thenY is proximinal with

x−PY(x)=x₀∗(x)D−1x₀∗ for anyx∈X.

Therefore,PY(x)is a singleton⇔D−1(x₀∗)is a singleton. Thus(b)⇒(a).

(b)Let x₀∗ be a τ-strongly support functional. Then kerx₀∗=Y is approximatively τ-compact in X. Suppose thatD−1is not norm-τusc atx₀∗. Then for someτ-open setWinXwithD−1(x₀∗)⊆W, there exists{x_n∗} ⊆SX∗such

thatx∗

n−x∗0 →0 andD−1(xn∗)Wfor alln.

Letzn∈D−1(xn∗)\W. Fixx0∈D−1(x0∗). Letxn=x₀∗(zn)x0−zn. Then{xn} ⊆Y is a minimizing sequence forx0.

SinceY approximativelyτ-compact,{xn}has aτ-convergent subsequence. So{zn}has aτ-convergent subsequence

converging to somez0. Thusz0∈D−1(x∗₀)⊆W, butzn∈X\WandX\Wisτ-closed, a contradiction. ✷

Remark 2.15.It is well known that strongly exposing functionals are precisely the points ofSX∗ at which the dual

norm is Fréchet differentiable. Is there an analogous characterization of strongly support functionals?

Notice that one of the natural candidates, namely, the points ofSX∗ at which the dual norm is strongly

subdif-ferentiable (SSD) is actually weaker. To see this, recall that the dual norm is SSD at everyx∗∈NA(X)⇔ every proximinal hyperplane is strongly proximinal [8, Proposition 2.6] and that this condition holds inc0. Compare this

with Theorem 2.11 in the light of Example 2.4.

We conclude this section with a result about whenPY is weak–weak continuous. In [13], the authors prove that

ifY is a finite-dimensional Chebyshev subspace ofX such thatP_Y−1(0)is weakly closed, thenPY is weak–weak

continuous. The following theorem improves this result.

Theorem 2.16.IfY is a reflexive Chebyshev subspace ofXsuch thatP_Y−1(0)is weakly closed, thenPY is weak–weak

continuous.

Proof. For anyx ∈X, we havex−PY(x)−0 =d(x, Y )=d(x−PY(x), Y ), so x−PY(x)∈PY−1(0). Now if

{xα} ⊆Xandxα→x weakly, then{xα}is norm-bounded, i.e.,xαC for someC >0. Sincexα−PY(xα) =

d(xα, Y )xαC,PY(xα)2C. SinceY is reflexive, we have a subnetPY(xγ)that converges weakly to some

z∈Y. Thenxγ −PY(xγ)→x−zweakly. Since PY−1(0)is weakly closed andxα−PY(xα)∈PY−1(0), we have

x−z∈P_Y−1(0), so thatx−z =d(x−z, Y )=d(x, Y ). Therefore,z∈PY(x). SinceY is a Chebyshev subspace

ofX,PY(x)is a singleton. Thus, all weakly convergent subnets ofPY(xα)weakly converge toz. It follows thatPY(xα)

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3. Proximinality inτ-ALUR Banach spaces

Definition 3.1. (See [2].) Recall that a Banach spaceX isτ-almost locally uniformly rotund (τ-ALUR) if for any

x∈SX,{xn} ⊆BXand{xm∗} ⊆BX∗, the condition

lim

m limn x

∗

m _x

n+x

=1 impliesτ-limnxn=x.

In the literature, the acronym ALUR is also used for average locally uniformly rotund which is known to be equivalent toXis strictly convex and has the KK property. We will use it in the above sense only. Theorem 3.4 below discusses the relation between these two notions.

In this section, we consider proximinality inτ-ALUR spaces.

From the results of [1, Proposition 4.4 and Corollary 4.6], it follows that a Banach spaceXisτ-ALUR⇔every

x∗∈NA(X)is aτ-strongly exposing functional. Thus, Theorem 2.11 gives a new characterization ofτ-ALUR spaces. As noted earlier, if a finite-codimensional subspaceY ofXis proximinal inX, thenY⊥⊆NA(X). We now show that the converse holds inτ-ALUR spaces. It is not difficult to show that the conditionY⊥⊆NA(X)forcesX/Y to be reflexive [14, Lemma 2.2]. Thus, our result improves [9, Corollary 3.4].

Theorem 3.2.LetXbe aτ-ALUR Banach space andY be a subspace such thatX/Y is reflexive. Then the following are equivalent:

(a) Y is proximinal inX;

(b) Y isτ-strongly proximinal inX; (c) Y isτ-strongly Chebyshev inX; (d) Y⊥⊆NA(X).

Proof. By Theorem 2.11,(a)⇔(b)⇔(c).

(a)⇒(d). SinceX/Y is reflexive, everyx∗∈Y⊥≃(X/Y )∗is norm attaining onX/Y. Since Y is proximinal inX,x∗∈NA(X). ThusY⊥⊆NA(X).

(d)⇒(a). LetY⊥⊆NA(X). SinceXisτ-ALUR, everyx∗∈Y⊥≃(X/Y )∗is in particular an exposing functional. SinceX/Y is reflexive andX/Y=(Y⊥)∗, it is strictly convex. LetQ:X→X/Y be the quotient map. Ift∈SX/Y,

there isx∗∈Y⊥withx∗ =x∗(t )=1 and sinceX/Y is strictly convex,{u∈SX/Y: x∗(u)= u =1} = {t}. Now

sincex∗∈Y⊥⊆NA(X), there isx∈X withx =x∗(x)=1, and thusQ(x)=t. HenceQ(BX)=BX/Y andY is

proximinal inX. ✷

Remark 3.3.The same proof shows that if Y is a subspace ofX such that any x∗∈Y⊥is aτ-strongly exposing functional, thenY isτ-strongly Chebyshev.

LetY be a finite-codimensional proximinal subspace ofX. SupposeY⊥has a basis consisting ofτ-strongly ex-posing functionals. Does it imply thatY isτ-strongly Chebyshev inX?

We now compare ALUR with some related convexity notions.

Theorem 3.4.For a Banach spaceX,Xis LUR⇒Xis ALUR⇒Xis strictly convex and has the KK property. And neither converse implication holds.

Proof. The fact that LUR⇒ALUR, but not conversely, has been noted in [2]. Indeed, LUR⇒ALUR is clear from the definitions. And as noted in [2], ifXis an infinite-dimensional Banach space with separable dual, then there exists an equivalent norm onXsuch thatXis ALUR but fails to be LUR.

Assume now thatXis ALUR. ThenXis clearly strictly convex. Let{xn}, x0⊆SXbe such thatxn→x0weakly.

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LetX=c0and letT :ℓ2→Xbe a weakly compact operator with dense range. Define x∗ =x∗

₁+

T∗x∗

₂.

By [11, Proposition III.2.11], this defines an equivalent dual norm onX∗such thatc0is anM-ideal inℓ∞. This

new norm onX∗is strictly convex and has the KK property. Now we show that there are points inNA(X∗)that are not even w∗-exposing forBX∗∗∗.

IndeedX, with the ||| · |||-norm, is an M-ideal inX∗∗. SoX∗∗∗ can be decomposed as X∗∗∗=X∗⊕1X⊥. By

Bishop–Phelps Theorem, there isx∗∗∗∈SX∗∗∗\(X∗∪X⊥) such thatx∗∗∗∈NA(X∗∗). Then there is x∗∗∈X∗∗

such thatx∗∗∗(x∗∗)= |||x∗∗_{||| =}_{1. Now, we can write}_x∗∗∗₌_x∗₊_z∗∗∗ _with_x∗₌₀_∈_X∗_,_z∗∗∗₌₀_∈_X⊥_{and 1}₌ |||x∗∗∗||| = |||x∗|||+|||z∗∗∗|||. It follows thatx∗∗(x∗)= |||x∗|||, so thatx∗∗∈NA(X∗), butx∗∗does not w∗-exposex∗/|||x∗|||

inBX∗∗∗. ✷

Remark 3.5.Note that the last part of the proof shows that ifX is not reflexive and anM-ideal inX∗∗, thenX∗

always fails to be wALUR.

We conclude this section with the following result.

Theorem 3.6.An ALUR Banach spaceXis reflexive⇔the intersection of any two proximinal hyperplanes is prox-iminal.

Proof. IfXis reflexive, then all subspaces are proximinal.

Conversely, letX be ALUR and suppose the intersection of any two proximinal hyperplanes is proximinal. Let

x∗, y∗∈NA(X). Then kerx∗and kery∗are proximinal hyperplanes inX. IfY =kerx∗∩kery∗is proximinal inX, thenY⊥⊆NA(X), and therefore,αx∗+βy∗∈NA(X) for allα, β∈_R. This impliesNA(X)is a linear subspace ofX∗. SinceXis ALUR and since the set of strongly exposing functionals form aGδset,NA(X)is a denseGδinX∗.

Thus, by the Baire Category Theorem, for everyx∗∈X∗, we have (x∗+NA(X))∩NA(X)= ∅. HenceNA(X)−

NA(X)=X∗. SinceNA(X)is a linear subspace ofX∗,NA(X)=X∗, which, by James’ theorem, implies thatX is reflexive. ✷

4. Stability results

In this section, we prove some stability results on approximativelyτ-compact (respectivelyτ-strongly Chebyshev) subspaces.

Theorem 4.1. Let {Xi: 1im} be a family of Banach spaces and Yi be a subspace of Xi, respectively, for

1im. ConsiderX=

ℓpXi andY =

ℓpYi,1p <∞. ThenY is approximativelyτ-compact(respectively τ-strongly Chebyshev) inX⇔Yi is approximativelyτ-compact(respectivelyτ-strongly Chebyshev)inXi for all

1im.

Proof. We will prove the result for strongly Chebyshev subspaces. It will be clear that the remaining cases are similar. Suppose Y is strongly Chebyshev in X. Fix 1i m. Let xi ∈ Xi such that d(xi, Yi)=1. Let x =

(0, . . . ,0, xi,0, . . . ,0). Then for any y ∈ Y, x −ypp =j=iyjp + xi −yip xi −yip. It follows

that d(x, Y )=1 and the nearest point y0 must be of the form y0=(0, . . . ,0, yi,0, . . . ,0) withyi nearest to xi

inYi. Thus,Yi is proximinal inXi.

To prove thatYi is strongly Chebyshev inXi, let{yn,i}be a minimizing sequence forxi. Letyn=(0, . . . ,0, yn,i,0,

. . . ,0). Then{yn}is a minimizing sequence forx. So{yn}converges inY and this implies that{yn,i}converges inYi

proving thatYi is strongly Chebyshev inXi for 1im.

Conversely, suppose for all 1in,Yiis strongly Chebyshev inXi. First we prove thatY is proximinal inX. Let

x=(xi)1in∈X. For every 1im, there existsyi∈Yisuch thatxi−yi =d(xi, Yi). Lety=(yi)1in∈Y.

Then for anyz=(zi)1in∈Y,x−zpp=ni=1xi−zipmi=1xi−yip= x−ypp. Thus,yis nearest tox.

Now we claim thatY is strongly Chebyshev. Letx=(xi)1im∈Xsuch thatd(x, Y )=1 and{yn= {yn,i}}be a

minimizing sequence forx. Clearly{yn,i}is minimizing sequence forYi for 1im. Hence it converges inYi for

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Corollary 4.2.LetΛbe an index set. For allλ∈Λ, letYλbe a subspace of a Banach spaceXλ. For1p <∞, let

ℓpXλandY=

ℓpYλ.

(a) Y is approximativelyτ-compact(respectivelyτ-strongly Chebyshev)inX⇒Yλis approximatively τ-compact

(respectivelyτ-strongly Chebyshev)inXλfor allλ∈Λ.

(b) Y is proximinal inX⇔Yλis proximinal inXλfor allλ∈Λ.

Proof. The proof of (a) and the “necessity” part of(b)and(c)is essentially contained in the proof of Theorem 4.1. For “sufficiency” in both (b) and (c), let x =(xλ)∈X. For every λ∈ Λ, there exists yλ ∈ Yλ such that

xλ−yλ =d(xλ, Yλ). Lety=(yλ). It actually suffices to note thaty∈Y. This follows from the simple observation

thatxλ−yλ =d(xλ, Yλ)xλandx=(xλ)∈X. ✷

Remark 4.3.The above theorem and corollary do not hold forp= ∞. Indeed ifXi=R,i=1,2,3, andY1=Y2=R, Y3= {0}. LetX be theℓ∞sum ofXi,i=1,2,3, andY be theℓ∞sum ofYi. ThenYi is strongly Chebyshev inXi

fori=1,2,3, butY is not strongly Chebyshev as it is not Chebyshev subspace inX.

Remark 4.4.LetX be a Banach space and letμbe a Lebesgue measure on[0,1]. In [10,12], it is proved thatf is a strongly exposed point of the unit ball of Lebesgue–Bochner function spaceLp(μ, X), 1< p <∞, if and only if

f is a unit vector andf (t )/f (t )is strongly exposed point of the unit ball ofXfor almost allt in the support off. It follows thatXis ALUR if and only ifLp(μ, X)is ALUR. AndX admits strongly Chebyshev hyperplanes if and only ifLp(μ, X)does.

Acknowledgments

We thank Professor V. Indumathi for suggesting the term “strongly Chebyshev.” Some of the results presented in this paper have been included in the fourth authors PhD thesis written under the supervision of Professor Gilles Godefroy. It is a pleasure to thank him. We also thank Dr. Sudipta Dutta for many fruitful discussions. We also thank the referee for comments and suggestions that improved the paper.

References

[1] Pradipta Bandyopadhyay, Da Huang, Bor-Luh Lin, Rotund points, nested sequence of balls and smoothness in Banach spaces, Comment. Math. Prace Mat. 44 (2004) 163–186.

[2] P. Bandyopadhyay, Da Huang, Bor-Luh Lin, S.L. Troyanski, Some generalizations of locally uniform rotundity, J. Math. Anal. Appl. 252 (2000) 906–916.

[3] A.L. Brown, A rotund reflexive space having a subspace of codimension two with a discontinuous metric projection, Michigan Math. J. 21 (1974) 145–151.

[4] Stefan Cobza¸s, Geometric properties of Banach spaces and the existence of nearest and farthest points, Abstr. Appl. Anal. 2005 (2005) 259–285.

[5] F. Deutsch, Existence of best approximations, J. Approx. Theory 28 (1980) 132–154.

[6] N.V. Efimov, S.B. Stechkin, Approximative compactness and Chebyshev sets, Soviet Math. Dokl. 2 (1961) 1226–1228. [7] Ky Fan, Irving Glicksberg, Some geometric properties of the spheres in a normed linear space, Duke Math. J. 25 (1958) 553–568. [8] G. Godefroy, V. Indumathi, Strong proximinality and polyhedral spaces, Rev. Mat. Complut. 14 (2001) 105–125.

[9] G. Godefroy, V. Indumathi, F. Lust-Piquard, Strong subdifferentiability of convex functionals and proximinality, J. Approx. Theory 116 (2002) 397–415.

[10] Peter Greim, Strongly exposed points in BochnerLp-spaces, Proc. Amer. Math. Soc. 88 (1983) 81–84.

[11] P. Harmand, D. Werner, W. Werner,M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math., vol. 1547, Springer-Verlag, Berlin, 1993.

[12] Zhibao Hu, Bor-Luh Lin, Strongly exposed points in Lebesgue–Bochner function spaces, Proc. Amer. Math. Soc. 120 (1994) 1159–1165. [13] Clifford A. Kottman, Bor-Luh Lin, The weak continuity of metric projections, Michigan Math. J. 17 (1970) 401–404.

[14] Darapaneni Narayana, T.S.S.R.K. Rao, Transitivity of proximinality and norm attaining functionals, Colloq. Math. 104 (2006) 1–19. [15] Ivan Singer, Best Approximations in Normed Linear Spaces by Elements of Linear Subspaces, Grundlehren Math. Wiss., vol. 171,

Springer-Verlag, New York, 1970.

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Theorem 2.5.LetKbe aτ-closed subset of a Banach spaceXandx0∈X\K.

(a) Kisτ-strongly Chebyshev forx0⇒the metric projectionPK is single-valued and norm-τ continuous atx0, i.e., ifxn→x0in norm andzn∈PK(xn), thenzn→z0=PK(x0)inτ.

(b) Kis approximativelyτ-compact forx0⇒the metric projectionPKis norm-τ upper semicontinuous(usc)atx0, i.e., for anyτ-open setW withPK(x0)⊆W, there existsε >0such thatPK(x)⊆W wheneverx−x0< ε.

Proof. (a)IfK isτ-strongly Chebyshev forx0, thenPK(x0)is a singleton, say{z0}. Observe that, ifxn→x0and zn∈PK(xn), then{zn}is a minimizing sequence forx0. Therefore,zn→z0inτ-topology.

(b)AssumeKis approximativelyτ-compact forx₀. Then by Theorem 2.2,Kisτ-strongly proximinal forx₀and

PK(x0)isτ-compact.

LetW be aτ-open set such thatPK(x0)⊆W. SincePK(x0)isτ-compact, there exists aτ-open neighbourhoodV

of 0 such thatPK(x0)+V ⊆W. SinceK isτ-strongly proximinal forx0, there existsδ >0 such thatPK(x0, δ)⊆ PK(x0)+V ⊆W. Clearly, forx−x0< δ/2,PK(x)⊆PK(x0, δ)⊆W. ✷

Remark 2.6.Letτ be a topology onXsuch thatBXisτ-closed. This corresponds to a “regular mode of convergence” that is “topological” [5]. Examples include, apart from the norm, weak and weak∗ _{topologies, the strong and weak}

Similarly, if we call the sequential version, approximative τ-sequential compactness, then a τ-closed set K is approximativelyτ-sequentially compact forx₀∗⇒K isτ-strongly proximinal forx₀∗ andPK(x∗₀)isτ-sequentially compact. We do not know if the converse hold in general. Still, aτ-closed setK is approximativelyτ-sequentially compact forx₀∗⇒the metric projectionPK is norm-τ usc forx₀∗. This follows from [5, Theorem 2.7] and gives an alternative proof of Theorem 2.5(b).

Whenτ is the norm topology, we have

Proposition 2.7.LetXbe a Banach space,Kbe a closed subset ofXandx₀∈X\K. ThenKis strongly Chebyshev forx₀⇔for everyε >0, there existsδ >0such that · -diam(PK(x0, δ)) < ε.

Proof. IfKis strongly Chebyshev forx0, thenPK(x0)= {z0}. Therefore every minimizing sequence converges toz0.

Now, if the given condition does not hold, there existsε >0 andzn∈PK(x0,1/n)such thatzn−z0ε. Then{zn} is a minimizing sequence that does not converge toz0. Contradiction!

Conversely, if for everyε >0, there existsδ >0 such that · -diam(PK(x0, δ)) < ε, then the Cantor Intersection

Theorem shows thatPK(x0)is a singleton{z0}(say). Suppose{zn} ⊆Kis a minimizing sequence forx0. Letε >0

and letδ >0 be such that · -diam(PK(x0, δ)) < ε. Then, for all sufficiently large n,zn∈PK(x0, δ). Therefore, zn−z0 · -diam(PK(x0, δ)) < ε. ✷

Fan and Glicksberg [7] showed that in a strictly convex reflexive spaceX with the Kadec–Klee (KK) property (i.e.,xn, x∈BX, limnxn = x =1 and w-limnxn=x⇒limnxn−x =0), the metric projectionPK is norm– norm continuous for all nonempty closed convex setsK⊆Xand they have given various characterizations for such spaces. The following result characterizes reflexive spaces from proximinality point of view. The proof follows from arguments in [5,7].

Theorem 2.8.LetXbe a Banach space.

(a) Every closed convex set is proximinal inX⇔Xis reflexive⇔every closed convex set is approximatively weakly compact inX.

(b) Every closed convex set is weakly strongly Chebyshev inX⇔Xis reflexive and strictly convex.

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A.L. Brown [3] has shown that there exists a separable strictly convex and reflexive real Banach spaceX and a closed subspaceY such thatPY is not norm–norm continuous.

We now characterize approximativelyτ-compact andτ-strongly Chebyshev hyperplanes. Definition 2.9.

(a) Recall that x∗ ∈SX∗ is called a τ-strongly exposing functional if x∗∈NA(X) and every {x_n} ⊆B_X with

limx∗(xn)=1 isτ-convergent.

(b) Let us callx∗∈SX∗aτ-strongly support functional ifx∗∈NA(X)and every{x_n} ⊆B_Xwith limx∗(x_n)=1 has

aτ-convergent subsequence.

Theorem 2.10.For a Banach spaceX andx∗∈SX∗,kerx∗ is approximativelyτ-compact(respectivelyτ-strongly Chebyshev)inX⇔x∗is aτ-strongly support(respectivelyτ-strongly exposing)functional.

Proof. SupposeY=kerx∗is approximativelyτ-compact. ThenY is proximinal, and hence,x∗∈NA(X). Thus, there existsx0∈SXsuch thatx∗(x0)=1. Let{xn} ⊆SXsuch thatx∗(xn)→1. Letyn=x∗(xn)x0−xn∈Y. Then

1=x∗(x0)=x∗(x0−yn)x0−yn =x₀−x∗(xn)x0+xn

1−x∗(xn)+1→1.

Thus,{yn}is a minimizing sequence forx0, and hence has aτ-convergent subsequence. It follows that so does{xn}. Conversely, suppose x∗ is a τ-strongly support functional. Then, x∗ ∈NA(X), and hence, Y =kerx∗ is a proximinal subspace. Indeed, let x0∈SX be such that x∗(x0)=1. Then for any x ∈X, d(x, Y )= |x∗(x)| and x−x∗(x)x0∈PY(x).

Let {zn} ⊆Y be a minimizing sequence for x∈X\Y. It follows that x−zn →d(x, Y )= |x∗(x)| =0 and

x∗(x−zn)=x∗(x). Letwn=ηx(x−zn)/x−zn, whereηx is the sign of x∗(x), i.e.,ηxx∗(x)= |x∗(x)|. Then

wn =1 and the sequence x∗(wn)→1. Sincex∗is aτ-strongly support functional, the sequence{wn}has aτ -convergent subsequence. And hence,{zn}has aτ-convergent subsequence too.

The other case is similar. ✷

The following is the analogue of Theorem 2.8 without reflexivity. Theorem 2.11.For a Banach spaceX, the following are equivalent:

(a) Everyx∗∈NA(X)is aτ-strongly support(respectivelyτ-strongly exposing)functional.

(b) Every proximinal convex setK⊆Xis approximativelyτ-compact(respectivelyτ-strongly Chebyshev).

(d) Every proximinal hyperplane is approximativelyτ-compact(respectivelyτ-strongly Chebyshev).

Proof. (a)⇒(b). Suppose everyx∗∈NA(X)is aτ-strongly support functional. LetK be a proximinal convex set inX andx0∈X\K. Without loss of generality, we may assumex0=0 andd(0, K)=1. Then there existsz0∈K

support functional. Let{zn} ⊆Kbe a minimizing sequence for 0. Then 1x∗(zn)zn →1. Therefore,{zn/zn}, and hence{zn}, has aτ-convergent subsequence.

(b)⇒(c)⇒(d)is trivial, and(d)⇒(a)follows from Theorem 2.10. ✷

Remark 2.12.Observe that a subspaceY need not be strongly Chebyshev even ifY is a Chebyshev subspace ofX

andPY is norm–norm continuous. For example, letx∗∈SX∗ be an exposing functional that is not strongly exposing.

Thenx∗attains its norm at auniquepointx0∈SX. Thus,Y=kerx∗is Chebyshev withPY(x)=x−x∗(x)x0for any x∈X, so thatPY is continuous. But sincex∗is not strongly exposing, by Theorem 2.10,Y is not strongly Chebyshev. We do not know if Theorem 2.11 has an analogue for a “regular mode of convergence,” “topological” or otherwise.

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Definition 2.13.Forx∗∈X∗\ {0}andx∈X, let us define the following maps:

D(x)=x∗∈SX∗: x∗(x)= x, D−1x∗=x∈SX: x∗(x)=x∗

Dis called the duality map andD−1is called the pre-duality map. Naturally,D−1is defined only onNA(X). Theorem 2.14.LetXbe a Banach space andx₀∗∈SX∗.

(a) x₀∗is aτ-strongly exposing functional⇒the pre-duality mapD−1is single-valued and norm-τ continuous atx₀∗, i.e., ifx_n∗→x₀∗in norm andxn∈D−1(xn∗), thenxn→x0=D−1(x0∗)inτ.

(b) x₀∗is aτ-strongly support functional⇒the pre-duality mapD−1is norm-τ usc atx₀∗, i.e., for anyτ-open setW withD−1(x₀∗)⊆W, there existsε >0such thatD−1(x∗)⊆W wheneverx∗−x₀∗< ε.

Proof. Observe that ifx₀∗∈NA(X)andY =kerx₀∗, thenY is proximinal with

x−PY(x)=x₀∗(x)D−1x₀∗ for anyx∈X.

Therefore,PY(x)is a singleton⇔D−1(x₀∗)is a singleton. Thus(b)⇒(a).

thatx∗

n−x∗0 →0 andD−1(xn∗)Wfor alln.

Letzn∈D−1(xn∗)\W. Fixx0∈D−1(x0∗). Letxn=x₀∗(zn)x0−zn. Then{xn} ⊆Y is a minimizing sequence forx0.

SinceY approximativelyτ-compact,{xn}has aτ-convergent subsequence. So{zn}has aτ-convergent subsequence converging to somez0. Thusz0∈D−1(x∗₀)⊆W, butzn∈X\WandX\Wisτ-closed, a contradiction. ✷ Remark 2.15.It is well known that strongly exposing functionals are precisely the points ofSX∗ at which the dual

norm is Fréchet differentiable. Is there an analogous characterization of strongly support functionals?

Notice that one of the natural candidates, namely, the points ofSX∗ at which the dual norm is strongly

subdif-ferentiable (SSD) is actually weaker. To see this, recall that the dual norm is SSD at everyx∗∈NA(X)⇔ every proximinal hyperplane is strongly proximinal [8, Proposition 2.6] and that this condition holds inc0. Compare this

with Theorem 2.11 in the light of Example 2.4.

We conclude this section with a result about whenPY is weak–weak continuous. In [13], the authors prove that ifY is a finite-dimensional Chebyshev subspace ofX such thatP_Y−1(0)is weakly closed, thenPY is weak–weak continuous. The following theorem improves this result.

Theorem 2.16.IfY is a reflexive Chebyshev subspace ofXsuch thatP_Y−1(0)is weakly closed, thenPY is weak–weak

continuous.

Proof. For anyx ∈X, we havex−PY(x)−0 =d(x, Y )=d(x−PY(x), Y ), so x−PY(x)∈PY−1(0). Now if

{xα} ⊆Xandxα→x weakly, then{xα}is norm-bounded, i.e.,xαC for someC >0. Sincexα−PY(xα) =

d(xα, Y )xαC,PY(xα)2C. SinceY is reflexive, we have a subnetPY(xγ)that converges weakly to some

z∈Y. Thenxγ −PY(xγ)→x−zweakly. Since PY−1(0)is weakly closed andxα−PY(xα)∈PY−1(0), we have

x−z∈P_Y−1(0), so thatx−z =d(x−z, Y )=d(x, Y ). Therefore,z∈PY(x). SinceY is a Chebyshev subspace ofX,PY(x)is a singleton. Thus, all weakly convergent subnets ofPY(xα)weakly converge toz. It follows thatPY(xα) weakly converges toPY(x). ✷

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3. Proximinality inτ-ALUR Banach spaces

Definition 3.1. (See [2].) Recall that a Banach spaceX isτ-almost locally uniformly rotund (τ-ALUR) if for any

x∈SX,{xn} ⊆BXand{xm∗} ⊆BX∗, the condition

lim m limn x

∗

n+x 2

=1 impliesτ-limnxn=x.

In this section, we consider proximinality inτ-ALUR spaces.

From the results of [1, Proposition 4.4 and Corollary 4.6], it follows that a Banach spaceXisτ-ALUR⇔every

x∗∈NA(X)is aτ-strongly exposing functional. Thus, Theorem 2.11 gives a new characterization ofτ-ALUR spaces. As noted earlier, if a finite-codimensional subspaceY ofXis proximinal inX, thenY⊥⊆NA(X). We now show that the converse holds inτ-ALUR spaces. It is not difficult to show that the conditionY⊥⊆NA(X)forcesX/Y to be reflexive [14, Lemma 2.2]. Thus, our result improves [9, Corollary 3.4].

Theorem 3.2.LetXbe aτ-ALUR Banach space andY be a subspace such thatX/Y is reflexive. Then the following are equivalent:

(a) Y is proximinal inX;

(b) Y isτ-strongly proximinal inX; (c) Y isτ-strongly Chebyshev inX; (d) Y⊥⊆NA(X).

Proof. By Theorem 2.11,(a)⇔(b)⇔(c).

(a)⇒(d). SinceX/Y is reflexive, everyx∗∈Y⊥≃(X/Y )∗is norm attaining onX/Y. Since Y is proximinal inX,x∗∈NA(X). ThusY⊥⊆NA(X).

(d)⇒(a). LetY⊥⊆NA(X). SinceXisτ-ALUR, everyx∗∈Y⊥≃(X/Y )∗is in particular an exposing functional. SinceX/Y is reflexive andX/Y=(Y⊥)∗, it is strictly convex. LetQ:X→X/Y be the quotient map. Ift∈SX/Y, there isx∗∈Y⊥withx∗ =x∗(t )=1 and sinceX/Y is strictly convex,{u∈SX/Y: x∗(u)= u =1} = {t}. Now sincex∗∈Y⊥⊆NA(X), there isx∈X withx =x∗(x)=1, and thusQ(x)=t. HenceQ(BX)=BX/Y andY is proximinal inX. ✷

Remark 3.3.The same proof shows that if Y is a subspace ofX such that any x∗∈Y⊥is aτ-strongly exposing functional, thenY isτ-strongly Chebyshev.

LetY be a finite-codimensional proximinal subspace ofX. SupposeY⊥has a basis consisting ofτ-strongly ex-posing functionals. Does it imply thatY isτ-strongly Chebyshev inX?

We now compare ALUR with some related convexity notions.

Theorem 3.4.For a Banach spaceX,Xis LUR⇒Xis ALUR⇒Xis strictly convex and has the KK property. And neither converse implication holds.

Assume now thatXis ALUR. ThenXis clearly strictly convex. Let{xn}, x0⊆SXbe such thatxn→x0weakly.

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LetX=c0and letT :ℓ2→Xbe a weakly compact operator with dense range. Define x∗ =x∗

₁+

T∗x∗

₂.

By [11, Proposition III.2.11], this defines an equivalent dual norm onX∗such thatc0is anM-ideal inℓ∞. This

new norm onX∗is strictly convex and has the KK property. Now we show that there are points inNA(X∗)that are not even w∗-exposing forBX∗∗∗.

IndeedX, with the ||| · |||-norm, is an M-ideal inX∗∗. SoX∗∗∗ can be decomposed as X∗∗∗=X∗⊕1X⊥. By

Bishop–Phelps Theorem, there isx∗∗∗∈SX∗∗∗\(X∗∪X⊥) such thatx∗∗∗∈NA(X∗∗). Then there is x∗∗∈X∗∗

such thatx∗∗∗(x∗∗)= |||x∗∗_{||| =}_{1. Now, we can write}_x∗∗∗₌_x∗₊_z∗∗∗ _with_x∗₌₀_∈_X∗_,_z∗∗∗₌₀_∈_X⊥_{and 1}₌

|||x∗∗∗||| = |||x∗|||+|||z∗∗∗|||. It follows thatx∗∗(x∗)= |||x∗|||, so thatx∗∗∈NA(X∗), butx∗∗does not w∗-exposex∗/|||x∗|||

inBX∗∗∗. ✷

Remark 3.5.Note that the last part of the proof shows that ifX is not reflexive and anM-ideal inX∗∗, thenX∗

always fails to be wALUR.

We conclude this section with the following result.

Theorem 3.6.An ALUR Banach spaceXis reflexive⇔the intersection of any two proximinal hyperplanes is prox-iminal.

Proof. IfXis reflexive, then all subspaces are proximinal.

Conversely, letX be ALUR and suppose the intersection of any two proximinal hyperplanes is proximinal. Let

x∗, y∗∈NA(X). Then kerx∗and kery∗are proximinal hyperplanes inX. IfY =kerx∗∩kery∗is proximinal inX, thenY⊥⊆NA(X), and therefore,αx∗+βy∗∈NA(X) for allα, β∈_R. This impliesNA(X)is a linear subspace ofX∗. SinceXis ALUR and since the set of strongly exposing functionals form aGδset,NA(X)is a denseGδinX∗. Thus, by the Baire Category Theorem, for everyx∗∈X∗, we have (x∗+NA(X))∩NA(X)= ∅. HenceNA(X)− NA(X)=X∗. SinceNA(X)is a linear subspace ofX∗,NA(X)=X∗, which, by James’ theorem, implies thatX is reflexive. ✷

4. Stability results

In this section, we prove some stability results on approximativelyτ-compact (respectivelyτ-strongly Chebyshev) subspaces.

Theorem 4.1. Let {Xi: 1im} be a family of Banach spaces and Yi be a subspace of Xi, respectively, for 1im. ConsiderX=

ℓpXi andY =

ℓpYi,1p <∞. ThenY is approximativelyτ-compact(respectively

τ-strongly Chebyshev) inX⇔Yi is approximativelyτ-compact(respectivelyτ-strongly Chebyshev)inXi for all 1im.

(0, . . . ,0, xi,0, . . . ,0). Then for any y ∈ Y, x −ypp =j=iyjp + xi −yip xi −yip. It follows that d(x, Y )=1 and the nearest point y0 must be of the form y0=(0, . . . ,0, yi,0, . . . ,0) withyi nearest to xi inYi. Thus,Yi is proximinal inXi.

To prove thatYi is strongly Chebyshev inXi, let{yn,i}be a minimizing sequence forxi. Letyn=(0, . . . ,0, yn,i,0,

. . . ,0). Then{yn}is a minimizing sequence forx. So{yn}converges inY and this implies that{yn,i}converges inYi proving thatYi is strongly Chebyshev inXi for 1im.

Conversely, suppose for all 1in,Yiis strongly Chebyshev inXi. First we prove thatY is proximinal inX. Let

x=(xi)1in∈X. For every 1im, there existsyi∈Yisuch thatxi−yi =d(xi, Yi). Lety=(yi)1in∈Y. Then for anyz=(zi)1in∈Y,x−zpp=ni=1xi−zipmi=1xi−yip= x−ypp. Thus,yis nearest tox. Now we claim thatY is strongly Chebyshev. Letx=(xi)1im∈Xsuch thatd(x, Y )=1 and{yn= {yn,i}}be a minimizing sequence forx. Clearly{yn,i}is minimizing sequence forYi for 1im. Hence it converges inYi for 1im. Now it is easy to see that{yn}is a converging minimizing sequence forx. ✷

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Corollary 4.2.LetΛbe an index set. For allλ∈Λ, letYλbe a subspace of a Banach spaceXλ. For1p <∞, let

ℓpXλandY=

ℓpYλ.

(a) Y is approximativelyτ-compact(respectivelyτ-strongly Chebyshev)inX⇒Yλis approximatively τ-compact (respectivelyτ-strongly Chebyshev)inXλfor allλ∈Λ.

(b) Y is proximinal inX⇔Yλis proximinal inXλfor allλ∈Λ. (c) Y is Chebyshev inX⇔Yλis Chebyshev inXλfor allλ∈Λ.

xλ−yλ =d(xλ, Yλ). Lety=(yλ). It actually suffices to note thaty∈Y. This follows from the simple observation thatxλ−yλ =d(xλ, Yλ)xλandx=(xλ)∈X. ✷