Convexities and Existence of the Farthest Point

Abstract

Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space 𝑋 is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of 𝑋 has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of 𝑋 when 𝑋 is very convex.

1. Introduction

Let 𝑋 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, respectively. Let 𝑥∈𝑆(𝑋),Σ(𝑥)={𝑥^∗∈𝑆(𝑋^∗)∶𝑥^∗(𝑥)=1}. For any sequence {𝑥_𝑛}⊂𝑋, define sep(𝑥_𝑛)≡inf{‖𝑥_𝑛−𝑥_𝑚‖∶𝑛≠𝑚}. Let 𝐵 be a bounded subset of 𝑋. We define a real-valued function 𝑟_𝐵∶𝑋→𝑅 by 𝑟_𝐵(𝑥)=sup𝑥−𝑦∶𝑦∈𝐵 (1.1) and call 𝑟_𝐵(𝑥) the farthest distance from 𝑥 to 𝐵. The function 𝑟_𝐵 is convex and Lipschitz-continuous. In fact, |𝑟_𝐵(𝑥)−𝑟_𝐵(𝑦)|≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝑋.

A point 𝑧∈𝐵 is called a farthest point of 𝐵 if there exists an 𝑥∈𝑋 such that ‖𝑥−𝑧‖=𝑟_𝐵(𝑥).

The mapping 𝐹_𝐵∶𝑋→2^𝐵 defined by 𝐹_𝐵(𝑥)={𝑧∈𝐵∶‖𝑥−𝑧‖=𝑟_𝐵(𝑥)} is called the farthest point map of 𝐵.

The existence of a farthest point of 𝐵 is equivalent to the fact that the set 𝐷=𝑥∈𝑋∶‖𝑥−𝑧‖=𝑟_𝐵(𝑥)forsome𝑧∈𝐵 (1.2) is nonempty. In [1–5], the existence of a farthest point of 𝐵 is studied. Edelstein [2] showed that if 𝑋 is uniformly convex space, then the set 𝐷 defined above is dense in 𝑋. Asplund [1] showed that if 𝑋 is both reflexive and locally uniformly rotund (LUR), then the set 𝐷 is dense in 𝑋. Lau [4] proved that if 𝐵 is weakly compact subset of 𝑋, then 𝐷 contains a dense 𝐺_𝛿 set of 𝑋, and if 𝑋 is reflexive Banach space, then for every bounded, weakly closed subset 𝐵 in 𝑋, 𝐷 contains a dense 𝐺_𝛿 subset of 𝑋.

We say that 𝑋 is strongly convex (resp., very convex/nearly strongly convex/nearly very convex) if any 𝑥∈𝑆(𝑋) and {𝑥_𝑛}⊂𝐵(𝑋) with 𝑥^∗(𝑥_𝑛)→1 as 𝑛→∞ for some 𝑥^∗∈Σ(𝑥) imply 𝑥_𝑛→𝑥 as 𝑛→∞ (resp., 𝑥−→_𝑛^𝑤𝑥 as 𝑛→∞/{𝑥_𝑛} is relatively compact/{𝑥_𝑛} is weakly relatively compact).

The author [6] proved that strong convexity (resp., very convex/nearly strong convexity/nearly very convex) has important applications in approximation theory. Bandyopadhyay et al. [7] also proposed two generalizations of LUR and weakly locally uniform rotundity (WLUR), which were called almost locally uniform rotundity (ALUR) and weakly almost locally uniform rotundity (WALUR). A Banach space 𝑋 is said to be ALUR (resp., WALUR) if for any 𝑥∈𝑆(𝑋),{𝑥_𝑛}⊂𝐵(𝑋) and {𝑥^∗_𝑚}⊂𝐵(𝑋^∗), the condition lim_𝑚lim_𝑛𝑥^∗_𝑚((𝑥_𝑛+𝑥)/2)=1 implies 𝑥_𝑛→𝑥 (reps., 𝑥−→_𝑛^𝑤𝑥). Recently, we have proved that ALUR and strong convexity, WALUR and very convex are equivalent, respectively [8]. Sullivan [9] defined very rotund space. A Banach space 𝑋 is known as very rotund if no 𝑥^∗∈𝑆(𝑋^∗) is simultaneously a norming element for some 𝑥∈𝑆(𝑋) and 𝑥^∗∗∈𝑆(𝑋^∗∗), where 𝑥≠𝑥^∗∗. The author [10] proved that very rotund space coincides with very convex space. By [6–12], we know that the four new convexities mentioned above have a lot of good properties and applications.

It is known that LUR, WLUR, midpoint locally uniform rotundity (MLUR) and weakly midpoint locally uniform rotundity (WMLUR) are four important convexities in the geometric theory of Banach spaces. By [9–12], the relation of the convexities mentioned above is shown in Figure 1 below.

Figure 1 

The relationship between the convexities.

The structure of this paper is as follows. In Section 2, We will give five counterexamples, which show the relations among the four new convexities mentioned above and LUR, WLUR, MLUR, WMLUR, and rotund (R).

In Section 3, we prove that if 𝑋 is nearly very convex space and if for every 𝑥∈𝑋, there exists 𝑥^∗₀∈𝜕𝑟_𝐵(𝑥) which attains its norm, then for every bounded, weakly closed subset 𝐵 of 𝑋, the set 𝐷 defined above contains a 𝐺_𝛿 subset of 𝑋, which improves the results of Edelstain [2], Asplund [1], and Lau [4]. Finally, we also prove a sufficient condition that the farthest point map 𝐹_𝐵 is single valued in a residual subset of 𝑋 when 𝑋 is very convex.

2. Some Counterexamples about Convexities

Lemma 2.1 (see [13]). Let {𝑥_𝛼∶𝛼∈𝐷} be a net in 𝑋 if for every 𝜀>0, there exists 𝛼_𝑠∈𝐷 such that the tail {𝑥_𝛼∶𝛼≥𝛼_𝑠} has a finite 𝜀-net, then {𝑥_𝛼} is a relatively compact subset in 𝑋.

Lemma 2.2. If 𝑋 is nearly strongly convex space, then 𝑋 has Kadec property, that is, if whenever 𝑥∈𝑆(𝑋) and {𝑥_𝛼} is a net in 𝑆(𝑋) such that 𝑥−→_𝛼^𝑤𝑥, then 𝑥_𝛼→𝑥. Particularly, 𝑋 has property 𝐻, that is, if whenever 𝑥∈𝑆(𝑋) and {𝑥_𝑛} is a sequence in 𝑆(𝑋) such that 𝑥−→_𝑛^𝑤𝑥, then 𝑥_𝑛→𝑥.

Proof. Suppose that net {𝑥_𝛼∶𝛼∈𝐷}⊂𝑆(𝑋),𝑥∈𝑆(𝑋) such that 𝑥−→_𝛼^𝑤𝑥, we will prove that 𝑥_𝛼→𝑥.Case 1. If for every 𝜀>0, there exists 𝛼_𝑠∈𝐷 such that {𝑥_𝛼∶𝛼≥𝛼_𝑠} has a finite 𝜀-net, by Lemma 2.1 and 𝑥−→_𝛼^𝑤𝑥, we may obtain that 𝑥_𝛼→𝑥.Case 2. If for every 𝜀₀>0 such that all tails of net {𝑥_𝛼∶𝛼∈𝐷} have no finite 𝜀₀-net, we take 𝑓∈𝑆(𝑋^∗) with 𝑓 in Σ(𝑥). Since 𝑥−→_𝛼^𝑤𝑥,𝑓(𝑥_𝛼)→𝑓(𝑥)=1. Choosing {𝛽_𝑛}⊂𝐷,𝛽₁≤𝛽₂≤⋯≤𝛽_𝑛≤⋯, we know that𝑓𝑥_𝛼1>1−𝑛, (2.1) for any 𝛼∈𝐷 with 𝛼≥𝛽_𝑛. Take 𝛼₁∈𝐷 such that 𝛼₁≥𝛽₁. For 𝑥_𝛼1 and 𝛽₂, we can choose 𝛼₂∈𝐷,𝛼₂≥𝛽₂ such that ‖𝑥_𝛼2−𝑥_𝛼1‖≥𝜀₀. Otherwise, {𝑥_𝛼∶𝛼≥𝛽₂}⊂{𝑥∈𝑋∶‖𝑥−𝑥_𝛼1‖<𝜀₀}, that is, the tail {𝑥_𝛼∶𝛼≥𝛽₂} has finite 𝜀₀-net. This is a contradiction with the assumption. For 𝑥_𝛼1,𝑥_𝛼2, and 𝛽₃, there exists 𝛼₃∈𝐷,𝛼₃≥𝛽₃ such that ‖𝑥_𝛼3−𝑥_𝛼1‖≥𝜀₀ and ‖𝑥_𝛼3−𝑥_𝛼2‖≥𝜀₀. Otherwise, {𝑥_𝛼∶𝛼≥𝛽₃}⊂{𝑥∈𝑋∶‖𝑥−𝑥_𝛼1‖<𝜀₀}∪{𝑥∈𝑋∶‖𝑥−𝑥_𝛼2‖<𝜀₀}, that is, the tail {𝑥_𝛼∶𝛼≥𝛽₃} has finite 𝜀₀-net. This is a contradiction with the assumption. According to the same method, we may choose a sequence {𝑥_𝛼𝑛}⊂{𝑥_𝛼∶𝛼∈𝐷} such that sep𝑥_𝛼𝑛=inf𝑥_𝛼𝑛−𝑥_𝛼𝑚∶𝑛≠𝑚≥𝜀₀,𝛼_𝑛≥𝛽_𝑛,𝑛=1,2,…. (2.2) Hence, we know that {𝑥_𝛼𝑛} is not relatively compact.
On the other hand, by (2.1), we have that1≥𝑓𝑥_𝛼𝑛1>1−𝑛. (2.3) This shows that 𝑓(𝑥_𝛼𝑛)→1. Since 𝑋 is nearly strongly convex, {𝑥_𝛼𝑛} is relatively compact which is a contradiction.

Example 2.3. There exists an MLUR space which is not a nearly strongly convex space.
Recall the equivalent norm defined on 𝑐₀ by Smith [14]. For 𝑘∈ℕ, define a mapping 𝑉_𝑘∶𝑐₀→ℝ¹,𝑉_𝑘(𝑥)=𝑉_𝑘(𝜉₁,𝜉₂,…)=sup{|𝜉₁−𝜉_𝑖|∶𝑖≥𝑘}. Let {𝛼_𝑛} be a sequence of positive real numbers, and ∑^∞_𝑛=2𝛼²_𝑛=1. Define two mappings 𝑉 and 𝑇 from 𝑐₀→𝑙₂ as follows:𝑉(𝑥)=𝛼₂𝑉₂(𝑥),𝛼₃𝑉₃(𝑥),…,𝑇(𝑥)=𝜉₁,𝛼₂𝜉₂,𝛼₃𝜉₃,…. (2.4) Since {𝑉_𝑘(𝑥)} and {𝜉_𝑘} are both bounded sequences, and ∑^∞_𝑛=2𝛼²_𝑛=1, then we know that {𝑉_𝑘(𝑥)} and 𝑇(𝑥) are in 𝑙₂. For 𝑥 in 𝑐₀, let ‖𝑥‖_𝐺=‖𝑥‖²_∞+‖𝑉(𝑥)‖²₂+‖𝑇(𝑥)‖²₂^1/2. (2.5) Since 𝑇 is one to one continuous, linear, and 𝑉(𝑥+𝑦)≤𝑉(𝑥)+𝑉(𝑦) for all 𝑥,𝑦 in 𝑐₀, the ‖⋅‖_𝐺 is a norm, and ‖⋅‖_∞≤‖⋅‖_𝐺≤√6‖⋅‖_∞. This shows that ‖⋅‖_𝐺 is an equivalent norm on 𝑐₀.
Smith shows that (𝑐₀,‖⋅‖_𝐺) is MLUR. We say that (𝑐₀,‖⋅‖_𝐺) has no property H. Indeed, let 𝑥=𝑒₁,𝑥_𝑛=𝑒₁+𝑒_𝑛, then ‖𝑥_𝑛‖_𝐺→‖𝑥‖_𝐺=√3 and 𝑥−→_𝑛^𝑤𝑥, but 𝑥_𝑛↛𝑥. By Lemma 2.2, we know that (𝑐₀,‖⋅‖_𝐺) is not nearly strongly convex.

Example 2.4. There exists a very convex space which is not nearly strongly convex space.
Recall the equivalent norm on Hilbert space by Troyanski in Isratescu [15]. Let 𝑋 be a Hilbert space, and let {𝑒_𝑖}^∞_𝑖=0 be an orthogonal basis. For any 𝑥∈𝑋,𝑥=𝜆₀𝑒₀+𝜆₁𝑒₁+⋯+𝜆_𝑛𝑒_𝑛+⋯, let‖𝑥‖²₁=max𝜆₀²+𝜆₂²+⋯+𝜆_2𝑛²+⋯,𝜆₁²+𝜆₃²+⋯+𝜆_2𝑛+1²+⋯. (2.6) It is obvious that ‖⋅‖₁ is an equivalent norm on 𝑋. Further, we set ‖𝑥‖=‖𝑥‖²₁+^∞_𝑖=112^𝑖𝜆_𝑖^21/2. (2.7) Clearly, this is again a norm on 𝑋 which is an equivalent original one. Troyanski shows that (𝑋,‖⋅‖) is R and reflexive [15], but it has no property H. Hence, (𝑋,‖⋅‖) is very convex, but is not nearly strongly convex by Lemma 2.2.

Example 2.5. There exists a nearly very convex space which is neither very convex space nor nearly strongly convex space.
For any 𝑥∈𝑙₂, let𝑥1|‖‖|=max√2‖𝑥‖₂,‖𝑥‖_∞. (2.8) Since (1/√2)‖𝑥‖₂≤|‖𝑥‖|≤‖𝑥‖₂,|‖𝑥‖| is an equivalent norm on 𝑙₂. Since (𝑙₂,|‖⋅‖|) is reflexive, we know that (𝑙₂,|‖⋅‖|) is nearly very convex space. We say that (𝑙₂,|‖⋅‖|) has no property H. Indeed, let 𝑥=𝑒₁,𝑥_𝑛=𝑒₁+𝑒_𝑛, then |‖𝑥‖|=|‖𝑥_𝑛‖|=1, but |‖𝑥_𝑛−𝑥_𝑚‖|=1. Hence, B (𝑙₂,|‖⋅‖|) is not nearly strongly convex by Lemma 2.2. We say that (𝑙₂,|‖⋅‖|) is not very convex. In fact, let 𝑒_1,2=(1,1,0,…),𝑒_1,3=(1,0,1,0,…), then |‖𝑒_1,2+𝑒_1,3‖|=2. This shows that (𝑙₂,|‖⋅‖|) is not R, and therefore, (𝑙₂,|‖⋅‖|) is not very convex.

Example 2.6. There exists a strongly convex space which is not WLUR space.
Let 𝐸=(𝑙₂,‖⋅‖), where 𝑥=(𝑎₁,𝑎₂,…)∈𝑙₂,‖𝑥‖²=𝑎₁+𝑎²₂+𝑎²₃+⋯^1/22+𝑎₂2²𝑎+⋯+^𝑛𝑛²+⋯. (2.9) Let 𝑋=(Σ⊕𝐸)_𝑙2. In [16], it is proved that 𝑋 is 2R, but is not KUR. Since 2R implies R, reflexive and property H, we get that 𝑋 is strongly convex space.
𝑋 is not WLUR space. Indeed, let {𝑒_𝑖}^∞_𝑖=1 be the natural basis, and 𝑥=(𝑒₁,𝑒₁,0,…),𝑥_𝑛=(1/2(𝑒₁+𝑒_𝑛),𝑒₁,0,…), then lim_𝑛→∞‖𝑥_𝑛‖=√2=‖𝑥‖ and lim_𝑛→∞‖𝑥_𝑛+𝑥‖=√8. However,𝑑𝑥_𝑛,span{𝑥}=inf𝑥_𝑛{𝑥≥1−𝑦∶𝑦∈span}2. (2.10) Choose 𝑓∈𝑆(𝑋^∗) such that 𝑓(𝑥_𝑛)≥1/2,𝑓(𝑦)=0,forall𝑦∈span{𝑥}. It is easily proved that lim_𝑛→∞𝑥‖𝑥‖+𝑥_𝑛𝑥_𝑛=1√2lim_𝑛→∞𝑥+𝑥_𝑛=2, (2.11) but 𝑓((𝑥/‖𝑥‖)+(𝑥_𝑛/‖𝑥_𝑛‖))≥|𝑓(𝑥_𝑛/‖𝑥_𝑛‖)|=1/(2√2). This shows that 𝑋 is not WLUR space.

Example 2.7. There exists a nearly strongly convex space which is not a strongly convex space.
For 𝑥=(𝑥₁,𝑥₂) in ℝ², let ‖𝑥‖=|𝑥₁|+|𝑥₂|, then (ℝ²,‖⋅‖) is reflexive and has property H. Hence, it is nearly strongly convex. It is easy to prove that (ℝ²,‖⋅‖) is not R space. Therefore (ℝ²,‖⋅‖) is not strongly convex.

Remark 2.8 (Smith [17]). gave three examples (𝑙₂,‖⋅‖_𝑊),(𝑙₂,‖⋅‖_𝐴),(𝑙₁,‖⋅‖_𝐻). He shows that (𝑙₂,‖⋅‖_𝑊) is WLUR not MLUR, and (𝑙₂,‖⋅‖_𝐴) is MLUR not WLUR, and (𝑙₁,‖⋅‖_𝐻) is R not MLUR. It is easily proved that (𝑙₁,‖⋅‖_𝐻) is not WMLUR either.

By the above five counterexamples and Remark 2.8, we know that, except for very convex implied WMLUR, none of the above converse implied relations in the diagram is generally true.

3. Convexities and Existence of the Farthest Point

Before proceeding to this part, let’s recall that the subdifferential of convex function 𝑓 on Banach space 𝑋 is defined by 𝜕𝑓(𝑥)=𝑥^∗∈𝑋^∗∶𝑥^∗𝑦−𝑥≤𝑓𝑦−𝑓(𝑥),∀𝑦∈𝑋. (3.1)

𝑥→𝜕𝑓(𝑥) is called subdifferential mapping.

Remark 3.1. It was shown in [4] that if 𝐵 is a bounded closed subset in Banach space 𝑋, then for any 𝑥∈𝑋 and 𝑥^∗∈𝜕𝑟_𝐵(𝑥), we have that ‖𝑥^∗‖≤1 and thus sup𝑥^∗(𝑥−𝑧)∶𝑧∈𝐵≤𝑥^∗⋅𝑟_𝐵(𝑥)≤𝑟_𝐵(𝑥). (3.2) Hence, for any 𝑥∈𝑋 and 𝑥^∗∈𝜕𝑟_𝐵(𝑥), we have that inf_𝑧∈𝐵𝑥^∗(𝑧−𝑥)≥−𝑟_𝐵(𝑥). (3.3)

Lemma 3.2 (Lau [4]). Let 𝑋 be a Banach space and 𝐵 a bounded subset in 𝑋, then the set 𝐹=𝑥∈𝑋∶inf_𝑧∈𝐵𝑥^∗(𝑧−𝑥)>−𝑟_𝐵(𝑥)forsome𝑥^∗∈𝜕𝑟_𝐵(𝑥) (3.4) is a first category in 𝑋.

Theorem 3.3. Let 𝑋 be a nearly very convex Banach space and 𝐵 a bounded, weakly closed subset of 𝑋. Further, for any 𝑥∈𝑋, if there exists an 𝑥^∗₀∈𝜕𝑟_𝐵(𝑥) which attains its norm, then 𝐷=𝑥∈𝑋∶‖𝑥−𝑧‖=𝑟_𝐵(𝑥)forsome𝑧∈𝐵 (3.5) contains a dense 𝐺_𝛿 set of 𝑋. In particular, the set of farthest points of 𝐵 is nonempty.

Proof. Define 𝐹 as in Lemma 3.2 and 𝐹_𝑛=𝑥∈𝐵∶inf_𝑧∈𝐵𝑥^∗(𝑧−𝑥)≥−𝑟_𝐵(𝑥)+1𝑛forsome𝑥^∗∈𝜕𝑟_𝐵(𝑥). (3.6) Let 𝑄=𝑋⧵𝐹, then 𝑄=𝑋⧵^∞_𝑛=1𝐹_𝑛=^∞_𝑛=1𝑋⧵𝐹_𝑛, (3.7) where each 𝑋⧵𝐹_𝑛 is an open, dense subset in 𝐵. Hence, 𝑄 is a dense 𝐺_𝛿 set in 𝑋.
Now, we prove 𝑄⊂𝐷 as follows. For any 𝑥∈𝑄, take 𝑥^∗₀∈𝜕𝑟_𝐵(𝑥) such that 𝑥^∗₀ attains its norm. Since inf_𝑧∈𝐵𝑥^∗₀(𝑧−𝑥)≥−𝑟_𝐵(𝑥), by the definition of 𝑄, we have that inf_𝑧∈𝐵𝑥^∗₀(𝑧−𝑥)≤−𝑟_𝐵(𝑥). Take sequence {𝑧_𝑛}⊂𝐵 such thatlim_𝑛→∞𝑥^∗₀𝑧_𝑛−𝑥=−𝑟_𝐵(𝑥). (3.8) Clearly, (𝑧_𝑛−𝑥)/(−𝑟_𝐵(𝑥))∈𝐵(𝑋). Given that 𝑋 is nearly very convex and that 𝐵 is weakly closed, there exist 𝑧₀∈𝐵 and subsequence {𝑧_𝑛𝑘}⊂{𝑧_𝑛} such that (𝑧_𝑛𝑘−𝑥)/(−𝑟_𝐵(𝑥))−→^𝑤(𝑧₀−𝑥)/(−𝑟_𝐵(𝑥)) as 𝑘→∞, that is, 𝑧−→_𝑛𝑘^𝑤𝑧₀ as 𝑘→∞. Hence, 𝑥^∗₀(𝑧₀−𝑥)=−𝑟_𝐵(𝑥). It follows that 𝑟_𝐵(𝑥)≥𝑥−𝑧₀≥𝑥^∗₀𝑧₀−𝑥=𝑟_𝐵(𝑥). (3.9) Thus, ‖𝑧₀−𝑥‖=𝑟_𝐵(𝑥). This shows that 𝑄⊂𝐷.

Corollary 3.4 (Lau [4]). If 𝑋 is a reflexive Banach space. Then for every bounded, weakly closed subset 𝐵 in 𝑋, the set 𝐷=𝑥∈𝑋∶‖𝑥−𝑧‖=𝑟_𝐵(𝑥)forsome𝑧∈𝐵 (3.10) contains a dense 𝐺_𝛿 set of 𝑋, and hence, the set of farthest points of 𝐵 is nonempty.

Corollary 3.5 (Asplund [1]). If 𝑋 is a reflexive LUR Banach space, then Corollary 3.4 holds for every bounded closed subset 𝐵 in 𝑋.

Theorem 3.6. Let 𝑋 be a very convex Banach space and 𝐵 a bounded, weakly closed subset of 𝑋. For any 𝑥∈𝑋, if there exists an 𝑥^∗₀∈𝜕𝑟_𝐵(𝑥) which attains its norm, then the farthest point map 𝐹_𝐵 is single valued in a residual subset of 𝑋.

Proof. By Theorem 3.3, 𝑄=𝑋⧵𝐹=𝑥∈𝑋∶sup_𝑧∈𝐵𝑥^∗(𝑥−𝑧)=𝑟_𝐵(𝑥)forany𝑥^∗∈𝜕𝑟_𝐵(𝑥) (3.11) is a dense 𝐺_𝛿 subset of 𝑋, where 𝐹 is defined as in Lemma 3.2.
Now we prove that 𝐹_𝐵(𝑥) is single valued for all 𝑥∈𝑄.
If 𝐹_𝐵 is not single valued on 𝑄, then there are 𝑥₀∈𝑄 and 𝑧₁,𝑧₂∈𝐵 with 𝑧₁≠𝑧₂ such that ‖𝑥₀−𝑧_𝑖‖=𝑟_𝐵(𝑥₀),𝑖=1,2. By Hahn-Banach theorem, we have 𝑥^∗_𝑖∈𝑆(𝑋^∗),𝑖=1,2 such that𝑥^∗_𝑖𝑥₀−𝑧_𝑖=𝑥₀−𝑧_𝑖,𝑖=1,2. (3.12) For any 𝑧^′∈𝐵,𝑦∈𝑋, we have that 𝑥^∗_𝑖𝑦−𝑥₀=𝑥^∗_𝑖𝑦−𝑧^′−𝑥^∗_𝑖𝑥₀−𝑧^′≤sup_𝑧∈𝐵sup_{𝑥∗(𝑋∈𝐵∗)}𝑥^∗𝑦−𝑧−𝑥^∗_𝑖𝑥₀−𝑧^′=𝑟_𝐵𝑦−𝑥^∗_𝑖𝑥₀−𝑧^′. (3.13) Thus, sup_𝑧∈𝐵𝑥^∗_𝑖𝑥₀−𝑧≤𝑟_𝐵𝑦−𝑥^∗_𝑖𝑦−𝑥₀. (3.14) Since 𝑥₀∈𝑄,𝑟_𝐵(𝑥₀)≤𝑟_𝐵(𝑦)−𝑥^∗_𝑖(𝑦−𝑥₀). This shows that 𝑥^∗_𝑖∈𝜕𝑟_𝐵(𝑥₀). Let 𝑥^∗₀=(1/2)𝑥^∗₁+(1/2)𝑥^∗₂, then 𝑥^∗₀∈𝜕𝑟_𝐵(𝑥₀) due to the convexity of 𝜕𝑟_𝐵(𝑥₀), and sup_𝑧∈𝐵𝑥^∗₀(𝑥₀−𝑧)=𝑟_𝐵(𝑥₀). Take a sequence {𝑧_𝑛}⊂𝐵 such that lim_𝑛→∞𝑥^∗₀𝑥₀−𝑧_𝑛=𝑟_𝐵𝑥₀. (3.15) It follows that lim_𝑛→∞𝑥^∗_𝑖(𝑥₀−𝑧_𝑛)=𝑟_𝐵(𝑥₀),𝑖=1,2. By (3.12), 𝑥^∗_𝑖∈Σ((𝑥₀−𝑧_𝑖)/(𝑟_𝐵(𝑥₀))),𝑖=1,2. Because 𝑋 is very convex, (𝑥₀−𝑧_𝑛)/(𝑟_𝐵(𝑥₀))−→^𝑤(𝑥₀−𝑧_𝑖)/(𝑟_𝐵(𝑥₀)),𝑖=1,2 as 𝑛→∞. According to uniqueness of weak limit point, we have that 𝑧₁=𝑧₂, which is a contradiction.

Remark 3.7. For closed-convex subset 𝐵 and bounded closed, relatively weakly compact 𝐾 in 𝑋, Ni and Li [18] proved that the set of all points in 𝐵 such that the farthest problem max{𝑥,𝐾} is well posed is a dense 𝐺_𝛿 subset in 𝐵 provided that 𝐵 is both strictly convex and Kadec with respect to 𝐾. This shows that the farthest point map 𝐹_𝐵 is single valued in a residual subset of 𝑋. By Example 2.4 in this paper, there exists a Banach space where assumptions in Theorem 3.6 are satisfied, but its unit ball 𝐵(𝑋) is not Kadec. Let 𝐵(𝑋)=𝐵=𝐾, then we know that conditions of Theorem 3.6 are different from conditions of the result by Ni and Li. Hence, the result by Ni and Li does not imply Theorem 3.6.

Acknowledgment

The authors would like to extend their heartfelt gratitude to those dear referees for their valuable suggestions.