# Convexity and Proximinality in Banach Space

Authors: Z. H. Zhang; C. Y. Liu
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

By the continuity of preduality map, we give some necessary and sufficient
conditions of the strongly convex and very convex spaces, respectively. Using
nearly strong convexity ofX, we give some equivalent conditions that every
element in X is strongly unique of order p, bounded strongly unique of order p, and locally strongly unique of order p.


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## Body

## 1. Notations and Definitions

LetX be a Banach space and let X* be its dual space. Let us denote by B(X) and S(X) the closed unit ball and the unit sphere of X, and by B(x,r) the closed ball centered at x of radius r>0. Let x∈S(X),D(x)={f∈S(X*):f(x)=1}. Let us denote by NA(X) the set of all norm-attaining functionals in X* and S0(X*)=S(X*)∩NA(X).For a subsetC⊂X, the metric projection PC:X→2C is defined by PC(x)={y∈C:∥x-y∥=d(x,C)}, where d(x,C)=inf{∥x-y∥:y∈C},x∈X. If PC(x)≠∅ for each x∈X, then C is said to be proximinal. If PC(x) is at most a singleton for each x∈X, then C is said to be semi-Chebyshev. If C is a simultaneously proximinal and semi-Chebyshev set, then C is said to be a Chebyshev set.Definition 1.1 (see [1]).
One says thatX is strongly convex (resp., very convex/nearly strongly convex/nearly very convex) if for any x∈S(X) and {xn}⊂B(X) convergence x*(xn)→1 as n→∞ for some x*∈D(x) implies that xn→x as n→∞ (resp., xn→wx as n→∞/{xn} is relatively compact/{xn} is weakly relatively compact).Definition 1.2 (see [2]).
One says thatX is midpoint locally uniformly rotund (MLUR) (resp., weakly midpoint locally uniformly rotund (WMLUR)) if any x0, xn, yn∈S(X), (n=1,2,…), xn+yn→2x0 as n→∞ implies that xn-yn→θ as n→∞ (resp., xn-yn→wθ as n→∞).Wu and Li defined strong convexity in [3], and Wang and Zhang in [4] defined very convexity, nearly strong convexity, and nearly very convexity which are two generalizations of locally uniformly rotund (LUR) and weakly locally uniformly rotund (WLUR) spaces. By [3, 5], we have the following relations:Sullivan defined very rotund space in [6]. A Banach space X is said to be very rotund if no x*∈S(X*) is simultaneously a norming element for some x∈S(X) and x**∈S(X**), where x≠x**. Z. H. Zhang and C. J. Zhang proved that very rotund space coincides with very convex space in [5]. In [3–5, 7], many results of these four classes of convexities were proved. In particular, Zhang and Shi proved that they have important applications in approximation theory in [1]. In [8], Bandyapadhyay et al. also proposed two generalizations of locally uniformly rotund space, which are called almost locally uniformly rotund space and weakly almost locally uniformly rotund space. A Banach space X is said to be ALUR (resp., WALUR) if for any x∈S(X), {xn}⊂B(X), and {xn}⊂B(X*), the condition limmlimnxm*((xn+x)/2)=1 implies xn→x (resp., xn→wx). Many properties of these two classes of convexities were studied in [8–10] too. Recently, we proved that almost locally uniformly rotund space is equivalent to strongly convex space and that weakly almost locally uniformly rotund space is equivalent to very convex space [7]. Thus, we unified the results of the studies about the strongly convex space (resp., very convex space) and the almost locally uniform rotundity (resp., weakly almost locally uniform rotundity). This shows that these convexities have important effects on and applications in geometry of Banach space and approximation theory.A sequence{zn}⊂C is said to be minimizing for x∈X∖C if ∥x-zn∥→d(x,C) as n→∞.Definition 1.3 (see [10]).
LetC be a closed subset (resp., a weakly closed subset) of X and x0∈X∖C.(1)
One says thatC is approximatively (resp., weakly approximatively) compact for x0 if every minimizing sequence {zn}⊂C for x0 has a convergent (resp., weakly convergent) subsequence.(2)
One says thatC is strongly Chebyshev (resp., weakly strongly Chebyshev) for x0 if every minimizing sequence {zn}⊂C for x0 is convergent (resp., weakly convergent).IfC is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) for every x∈X∖C, we say that C is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) in X.Definition 1.4 (see [11]).
LetG∈X, x∈X∖G, and 1≤p<∞.(1)
g0∈PG(x) is said to be strongly unique at x if there exists a constant rp>0 such that
(1.1)‖x-g‖≥‖x-g0‖+rp‖g-g0‖,
for any g∈G.(2)
g0∈G is said to be strongly unique of order p at x if there exists an rp=rp(x)>0 such that
(1.2)‖x-g‖p≥‖x-g0‖p+rp‖g-g0‖p,
for any g∈G.(3)
g0∈G is said to be bounded strongly unique of order p at x if given any N>0, there exists an rp,N=rp,N(x) such that
(1.3)‖x-g‖≥‖x-g0‖+rp,N‖g-g0‖p,
for any g∈G,∥g-g0∥≤N.(4)
g0∈G is said to be locally strongly unique of order p at x if there exist N>0 and rp,N=rp,N(x) such that
(1.4)‖x-g‖≥‖x-g0‖+rp,N‖g-g0‖p,
for any g∈G,∥g-g0∥≤N.In order to study the uniqueness of best approximation in nonlinear approximation theory, Wulbert [12] defined the strong uniqueness of best approximation. Smarzewski [13] and Schmidt [14] defined the strongly unique of order p and the bounded strongly unique of order p, respectively. By [11], we know that the strongly unique of order p and the bounded strongly unique of order p all are generalizations of the strongly unique. The strongly unique of order p and the bounded strongly unique of order p imply the locally strongly unique of order p, but converse implied relation is not generally true. When p=1, the strongly unique of order p and the bounded strongly unique of order p all are strongly unique. The locally strongly unique of order 1 is not equivalent to the strongly unique.Definition 1.5 (see [10]).
Forx*∈S(X*) and x∈S(X), let one define the following maps:
(1.5)D(x)={x*∈S(X*):x*(x)=1},D-1(x*)={x∈S(X):x*(x)=1}.D is called the duality map and D-1 is called the preduality map. Naturally, D-1 is defined only on S0(X*).Definition 1.6 (see [10]).
The preduality mapD-1:S(X*)→S(X) is said to be upper semicontinuous (n-τ) on S0(X*) if for any x*∈S0(X*) and any τ open set W with D-1(x*)⊂W, there exists ɛ>0 such that D-1(y*)⊂W whenever ∥x*-y*∥<ɛ, where τ stands for norm or weak topology.


## 2. Convexity and Continuity of the Preduality Map

Using the Bronsted-Rockafeller Theorem (see [15, Theorem  3.18, page 51]), we can prove the following lemma.Lemma 2.1.
Suppose thatɛ>0, x0∈S(X), x0*∈S(X*), and x0*(x0)>1-ɛ, then there are xɛ∈S(X) and xɛ*∈D(xɛ) such that
(2.1)‖xɛ-x0‖<2ɛ,‖xɛ*-x0*‖<2ɛ.Theorem 2.2.
LetX be a Banach space. X is strongly convex if and only if the preduality map D-1:S0(X*)→S(X) is singlevalued and continuous.Proof.
Necessity. Since strong convexity implies strict convexity, D-1 is singlevalued. Suppose that {xn*}⊂S0(X*), x*∈S0(X*) with xn*→x*, then there exist {xn}⊂S(X) and x∈S(X) such that xn*(xn)=1=x*(x). We have that
(2.2)x*(D-1(xn*))=(x*-xn*+xn*)(D-1(xn*))=(x*-xn*)(D-1(xn*))+xn*(D-1(xn*)).
Since
(2.3)|(x*-xn*)D-1(xn*)|≤‖xn*-x*‖⟶0,asn⟶∞,
and xn*(D-1(xn*))=xn*(xn)=1, we have that
(2.4)x*(D-1(xn*))=x*(xn)⟶1,asn⟶∞.
Since X is strongly convex, we deduce that xn→x as n→∞, that is, D-1(xn*)→D-1(x*) as n→∞. This means that D-1 is continuous.
Sufficiency. Let {xn}⊂S(X),x∈S(X) with x*(xn)→1 as n→∞ for some x*∈D(x). Since D-1 is singlevalued, by Lemma 2.1, there exist {yn}⊂S(X) and {yn*}⊂S0(X*) such that D-1(yn*)=yn and
(2.5)‖yn-xn‖⟶0,‖yn*-x*‖⟶0,asn⟶∞.
Since D-1 is continuous, we have that D-1(yn*)→D-1(x*) as n→∞, that is, yn→x as n→∞. So xn→x as n→∞, which means that X is strongly convex.Using Lemma2.1, in a similar way to prove Theorem 2.2, we can prove the following result.Theorem 2.3.
LetX be a Banach space. X is very convex if and only if the preduality map D-1:S0(X*)→S(X) is singlevalued and weakly continuous.Lemma 2.4.
LetC be a convex set of a strongly convex Banach space X. The following are equivalent:(1)
C is proximinal;(2)
C is weakly approximatively compact;(3)
C is approximatively compact;(4)
C is strongly Chebyshev.Proof.
We only need to prove(1)⇒(4).
Letx∈X∖C, {zn}⊂C such that
(2.6)lim⁡n→∞‖x-zn‖=d(x,C).
In order to finish the proof, we will show that there exists some y0∈C such that ∥y0-zn∥→0 as n→∞.
Step 1.
Ifx=0, then r=d(0,C)>0 and
(2.7)B(0,r)∩C=PC(0)≠∅,int⁡B(0,r)∩C=∅,
where B(0,r)={y:∥y∥≤r}. By the separation theorem [2] and definition of norm, there exists an f∈S(X*) such that
(2.8)sup⁡{f(y):y∈C}≤inf⁡{f(y):y∈B(0,r)}=-‖f‖r=-‖y0‖,
for any y0∈PC(0). Hence, we have that
(2.9)-‖y0‖≤f(y0)≤sup⁡{f(y):y∈C}≤-‖y0‖.
This shows that f∈D(-y0/∥y0∥). From zn∈C, we get f(zn)≤f(y0). Combining it with the condition limn→∞∥0-zn∥=d(0,C), it follows that
(2.10)‖0-y0‖=f(0-y0)≤f(0-zn)≤‖0-zn‖⟶d(0,C)=‖0-y0‖,
as n→∞. Hence, f(-zn/∥zn∥)→1 as n→∞. Since X is strongly convex, -zn/∥zn∥→-y0/∥y0∥ as n→∞, that is, ∥zn-y0∥→0 as n→∞.Step 2.
Ifx≠0, we set C′=x-C. It is clear that C′ is proximinal and {x-zn}⊂C′ is a minimizing sequence for 0. By Step 1, there exists y0′∈C′, such that ∥y0′-(x-zn)∥→0 as n→∞. This shows that ∥(x-y0′)-zn∥→0 as n→∞ and x-y0′=y0∈C.Similarly to the proof of Lemma2.4, we can prove the following result.Lemma 2.5.
LetC be a convex set of a nearly strongly convex Banach space X. The following are equivalent:(1)
C is proximinal;(2)
C is weakly approximatively compact;(3)
C is approximatively compact.Lemma 2.6.
Letx∈S(X), f∈D(x), then hyperplane H={y∈X:f(y)=1} is a proximinal convex subset in X.Proof.
Let(2.11)ker⁡f={y∈X:f(y)=0}.
We will prove that kerf is a proximinal convex subset in X. For any z∈X, if d(z,kerf)=0, since kerf is a closed subspace, we know that z∈kerf. Hence, z∈Pkerf(z). If d(z,kerf)>0, since X={αz}+kerf, there exist λ∈ℝ and y0∈kerf such that d(z,kerf)x=λz+y0. For any y∈X, d(y,kerf)=|f(y)|, and f(x)=1, we have that d(x,kerf)=1 and f(d(z,kerf)x)=d(d(z,kerf)x,kerf). Therefore, we have that
(2.12)d(z,ker⁡f)=d(λz+y0,ker⁡f)=|λ|d(z,ker⁡f).
This means that |λ|=1. Hence, we have
(2.13)d(z,ker⁡f)=‖d(z,ker⁡f)x‖=‖λz+y0‖=‖z+λy0‖.
It follows that -λy0∈Pkerf(z), which means that kerf is proximinal set.
Furthermore, we will prove that hyperplaneH={y∈X:f(y)=1} is a proximinal convex subset in X. For x0∈H, H=x0+kerf. For any z∈X,
(2.14)d(z,H)=inf⁡y∈ker⁡f‖z-(x0+y)‖=d(z-x0,ker⁡f).
Since kerf is a proximinal subset, there exists a y0∈kerf such that
(2.15)d(z-x0,ker⁡f)=‖z-(x0+y0)‖.
Therefore, x0+y0∈PH(z), which means that H is proximinal set in X.Theorem 2.7.
LetX be a Banach space. X is a nearly strongly convex if and only if the preduality map D-1:S0(X*)→S(X) is upper semicontinuous (n-n) on S0(X*) with norm compact images, where n stands for norm topology.Proof.
Necessity. Arbitrarily take x*∈S0(X*). Then by Lemma 2.6, we know that kerx*=Y is a proximinal convex subset in X. Since X is nearly strongly convex, by Lemma 2.5, we know that Y is approximatively compact. Suppose that D-1 is not upper semicontinuous (n-n) at x*, then for some open set W in X with D-1(x*)⊂W, there exists {xn*}⊂S(X*) such that xn*→x* as n→∞ and D-1(xn*)⊈W for all n. Let zn∈D-1(xn*)∖W. Fix x∈D-1(x*). Let xn=x*(zn)x-zn, then {xn}⊂Y is a minimizing sequence for x. Since Y is approximatively compact, {xn} has a convergent subsequence. So {zn} has convergent subsequence converging to z. Thus, z∈D-1(x*)⊂W, but zn∈X∖W is closed, which is a contradiction. By the assumption, we easily know that the image of D-1 is compact.
Sufficiency. Let {xn}⊂S(X),x∈S(X) with x*(xn)→1 as n→∞ for some x*∈D(x). By Lemma 2.1, there exist {yn}⊂S(X) and {yn*}⊂S(X*) such that yn∈D-1(yn*) and
(2.16)‖yn-xn‖⟶0,‖yn*-x*‖⟶0,asn⟶∞.
Since D-1(x*) is compact, D-1(x*) is proximinal. Let zn∈D-1(x*) such that ∥yn-zn∥=d(yn,D-1(x*)). In virtue of D-1 being upper semicontinuous (n-n) on S0(X*) and ∥yn*-x*∥→0, for any ɛ>0, there is n0 such that for all n≥n0d(yn,D-1(x*))<ɛ,thatis,∥yn-zn∥<ɛ, which means that yn-zn→θ as n→∞. Combining this with the compactness of D-1(x*),{zn} has convergent subsequence, and hence {yn} has convergent subsequence. By yn-xn→θ as n→∞, {xn} has convergent subsequence, which means that X is nearly strongly convex.Theorem 2.8.
LetX be a Banach space. X is nearly very convex if and only if the preduality map D-1:S0(X*)→S(X) is upper semicontinuous (n-w) on S0(X*) with weakly compact images.Proof.
In a similar way to the proof of Theorem2.7, the necessity can be proved.
Sufficiency. Let {xn}⊂S(X) with x*(xn)→1 as n→∞ for some x*∈D(x). By Lemma 2.1, there are {yn}⊂S(X) and {yn*}⊂S(X*) such that yn∈D-1(yn*) and
(2.17)‖yn-xn‖⟶0,‖yn*-x*‖⟶0asn⟶∞.
Suppose that {xn} does not have weakly convergent subsequence. By (2.17), we know that {yn} does not have weakly convergent subsequence. Without loss of generality, we can assume that
(2.18)|f(yn-ym)|≥ɛ0(∀n,m∈N,n≠m),
for some f∈X*∖{θ} and ɛ0>0. Since D-1(x*) is weakly compact, there exists {zi}i=1k⊂D-1(x*), such that D-1(x*)⊂∪i=1k{y∈X:|f(y-zi)|<ɛ0/2}. Combining upper semicontinuity (n-w) of D-1 with ∥yn*-x*∥→0,{yn}n≥n0⊂∪i=1k{y∈X:|f(y-zi)|<ɛ0/2} for some n0∈N. So, there are subsequence {ynk}k≥1⊂{yn}n≥n0 and i0(1≤i0≤k) such that {ynk}k≥1⊂{y∈X:|f(y-zi0)|<ɛ0/2}. Hence,
(2.19)|f(ynk-ynl)|≤|f(ynk-zi0)|+|f(zi0-ynl)|<ɛ02+ɛ02=ɛ0,
a contradiction with (2.18).


## 3. Convexity and Proximinality

Theorem 3.1.
LetC be a weakly approximatively compact subset of nearly strongly convex Banach space X, x∈X∖C, and g0∈C. If g0 is the unique element of best approximation of x, then the following are equivalent:(1)
g0 is strongly unique of order p at x;(2)
g0 is bounded strongly unique of order p at x;(3)
g0 is locally strongly unique of order p at x.Proof.
(1)⇒(2). Let rp>0 be such that
(3.1)‖x-g‖p≥‖x-g0‖p+rp‖g-g0‖p,
for any g∈C. For any N>0 and each g∈C,∥g-g0∥≤N, since
(3.2)‖x-g‖≤‖x-g0‖+‖g-g0‖≤‖x-g0‖+N,
by Lagrange’s mean value theorem,
(3.3)‖x-g‖p-‖x-g0‖p‖x-g‖-‖x-g0‖≤p(‖x-g0‖+N)p-1.
Set
(3.4)rp,N=rpp(‖x-g0‖+N)1-p.
Then
(3.5)‖x-g‖-‖x-g0‖≥rp,N‖g-g0‖p.
Because rp,N is independent of g,g0 is bounded strongly unique of order p at x.
(2)⇒(1). From
(3.6)lim⁡‖g‖→∞⁡‖x-g‖p-‖x-g0‖p‖g-g0‖p≥lim⁡‖g‖→∞⁡[(1-‖x-g0‖‖g-g0‖)p-(‖x-g0‖‖g-g0‖)p]=1,
we can take N>0 such that
(3.7)‖x-g‖p≥‖x-g0‖p+12‖g-g0‖p,∀g∈C,‖g-g0‖≥N.
Again, by Lagrange’s mean value theorem, we have
(3.8)‖x-g‖p-‖x-g0‖p‖x-g‖-‖x-g0‖≥p‖x-g0‖p-1.
Therefore,
(3.9)‖x-g‖p≥‖x-g0‖p+p‖x-g0‖p-1⋅[‖x-g‖-‖x-g0‖]≥‖x-g0‖p+p‖x-g0‖p-1⋅rp,N‖g-g0‖p,
for any g∈C and ∥g-g0∥≤N. Set rp=min{1/2,p∥x-g0∥p-1rp,N}, then
(3.10)‖x-g‖p≥‖x-g0‖p+rp‖g-g0‖p,
for any g∈C. That shows g0 is strongly unique of order p at x.
Now, we only need to prove(3)⇒(2). Let N0>0,rp,N0>0 such that
(3.11)‖x-g‖≥‖x-g0‖+rp,N0‖g-g0‖p,
for all g∈C and ∥g-g0∥≤N0.
If the condition(2) is not true, there exist N>N0, gn∈C with N0<∥gn-g∥≤N such that
(3.12)‖x-gn‖<‖x-g0‖+1n‖gn-g0‖p,
then, because {gn} is bounded,
(3.13)‖x-g0‖≤lim⁡n‖x-gn‖≤lim⁡n[‖x-g0‖+1n‖gn-g0‖p]=‖x-g0‖,
that is, limn∥x-gn∥=d(x,C). This shows that {gn} is a minimizing sequence for x. Since C is weakly approximatively compact, there is a weakly convergent subsequence of {gn}. Without loss of generality, we can assume that
(3.14)gn→wg0¯∈C.
Based on weak lower semi-continuity of norm, we get that
(3.15)‖x-g0¯‖≤lim⁡n‖x-gn‖=‖x-g0‖.
Hence, g0¯∈PC(x), consequently, g0¯=g0. Take f∈D((x-g0¯)/∥x-g0¯∥), then
(3.16)f(x-gn‖x-gn‖)⟶f(x-g0¯‖x-g0¯‖)=1.
Since X is nearly strongly convex, there exists {x-gnk}⊂{x-gn} such that x-gnk→x-g0¯, that is, gnk→g0¯. It follows that ∥g0¯-g0∥≥N0. This is a contradiction with the fact that g0¯=g0.Corollary 3.2.
LetC be a weakly approximatively compact subset of strongly convex Banach space X, x∈X∖C, and g0∈PC(x), then the following are equivalent:(1)
g0 is strongly unique of order p at x;(2)
g0 is bounded strongly unique of order p at x;(3)
g0 is locally strongly unique of order p at x.IfC is a convex set, we have the following result.Theorem 3.3.
LetC be a convex subset of a nearly strongly convex Banach space X, x∈X∖C, and g0∈C. If g0 is the unique element of best approximation of x, then the following are equivalent:(1)
g0 is strongly unique of order p at x;(2)
g0 is bounded strongly unique of order p at x;(3)
g0 is locally strongly unique of order p at x.Proof.
By the proof of Theorem3.1, we have (1)⇔(2). Now, we only need to prove (3)⇒(2).
If the condition(2) is not true, there exist N>N0,gn∈C with N0<∥gn-g∥≤N such that
(3.17)‖x-gn‖<‖x-g0‖+1n‖gn-g0‖p.
In the same way of the proof of (3)⇒(2) in Theorem 3.1, we can also prove that {gn} is a minimizing sequence for x. By Lemma 2.5, C is approximatively compact. Hence, there exists a convergent subsequence of {gn}. Without loss of generality, we can assume that
(3.18)gn⟶g0¯∈C.
Consequently, g0¯=g0, but ∥g0¯-g0∥≥N0, which is a contradiction.Corollary 3.4.
LetC be a convex set of a strongly convex Banach space X. x∈X∖C and g0∈PC(x). The following are equivalent:(1)
g0 is strongly unique of order p at x;(2)
g0 is bounded strongly unique of order p at x;(3)
g0 is locally strongly unique of order p at x.



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