Farthest Points and Subdifferential in 𝑝-Normed Spaces

Abstract

We study the farthest point mapping in a 𝑝-normed space 𝑋 in virtue of subdifferential of 𝑟(𝑥)=sup{‖𝑥−𝑧‖^𝑝∶𝑧∈𝑀}, where 𝑀 is a weakly sequentially compact subset of 𝑋. We show that the set of all points in 𝑋 which have farthest point in 𝑀 contains a dense 𝐺_𝛿 subset of 𝑋.

1. Introduction

Let 𝑋 be a real linear space. A quasinorm is a real-valued function on 𝑋 satisfying the following conditions. (i)‖𝑥‖≥0 for all 𝑥∈𝑋, and ‖𝑥‖=0 if and only if 𝑥=0.(ii)‖𝜆𝑥‖=|𝜆|‖𝑥‖ for all 𝜆∈𝑅 and all 𝑥∈𝑋.(iii)There is a constant 𝐾≥1 such that ‖𝑥+𝑦‖≤𝐾(‖𝑥‖+‖𝑦‖) for all 𝑥,𝑦∈𝑋. The pair (𝑋,‖⋅‖) is called a quasinormed space if ‖⋅‖ is a quasinorm on 𝑋. The smallest possible 𝐾 is called the modules of concavity of ‖⋅‖. By a quasi-Banach space we mean a complete quasinormed space, that is, a quasinormed space in which every Cauchy sequence converges in 𝑋.

This class includes Banach spaces. The most significant class of quasi-Banach spaces, which are not Banach spaces, is 𝐿_𝑝-spaces for 0<𝑝<1 equipped with the 𝐿_𝑝-norms ‖⋅‖_𝑝. A quasinorm ‖⋅‖ is called a 𝑝-norm (0<𝑝<1) if ‖𝑥+𝑦‖^𝑝≤‖𝑥‖^𝑝+‖𝑦‖^𝑝 for all 𝑥,𝑦∈𝑋. In this case, a quasinormed (quasi-Banach) space is called a 𝑝-normed (𝑝-Banach) space. By the Aoki-Rolewicz theorem [1], each quasinorm is equivalent to some 𝑝-norm. Since it is much easier to work with 𝑝-norms than with quasinorms, henceforth we restrict our attention mainly to 𝑝-norms. See [2–4] for more information.

If x* is in X*, the dual of 𝑋, and 𝑥∈𝑋 we write 𝑥^∗(𝑥) as ⟨𝑥^∗,𝑥⟩. We also consider quasinorms with 𝐾>1. The case where 𝐾=1 turns out to be the classical normed spaces, so we will not discuss it and refer the interested reader to [5–7] for analogue results concerning normed spaces.

In this paper, using some strategies from [5–7], we study the farthest point mapping in a 𝑝-normed space 𝑋 in virtue of subdifferential of 𝑟(𝑥)=sup{‖𝑥−𝑧‖^𝑝∶𝑧∈𝑀}, where 𝑀 is a weakly sequentially compact subset of 𝑋. We show that the set of all points in 𝑋 which have farthest point in 𝑀 contains a dense 𝐺_𝛿 subset of 𝑋.

Let 𝑋 be a 𝑝-normed space and let 𝑀 be a nonempty bounded subset of 𝑋. The mapping 𝑄_𝑀∶𝑋→2^𝑀 defined by 𝑄_𝑀(𝑥)={𝑧∈𝑀∶‖𝑥−𝑧‖^𝑝=sup_𝑡∈𝑀‖𝑥−𝑡‖^𝑝} is called the farthest point map of 𝑀. We call 𝑀 a remotal (uniquely remotal, resp.) set if for each 𝑥∈𝑋 the set 𝑄_𝑀(𝑥) is not empty (is singleton, resp.) [8–10].

2. Main Results

Let 𝑋 be a 𝑝-normed space and let 𝑀 be a bounded subset of 𝑋. For each 𝑥∈𝑋, we define the subdifferential of a function 𝑓 at 𝑥 by 𝜕𝑓(𝑥)={𝑥^∗∈𝑋^∗∶sgn⟨𝑥^∗,𝑦−𝑥⟩|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝+𝑓(𝑥)≤𝑓(𝑦)∀𝑦∈𝑋}. (2.1)This set may be empty even if we consider 𝑋 to be a Banach space [7, Example 3.8]. In a 𝑝-normed space, it may happen that 𝜕𝑟(𝑥)≠∅, although we should note that a 𝑝-normed space may have a trivial dual as well as it might have a nontrivial dual, see [11, Chapter 3], for some examples. To see the nonemptiness, suppose that 𝑋 is a 𝑝-normed space, 𝑥∈𝑋, and 𝑀={𝑥}. Thus, 𝑟(𝑥)=0 and obviously 0∈𝜕𝑟(𝑥). Also for each 𝑥^∗∈𝑋^∗ with ‖𝑥^∗‖≤1, we have|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝≤‖𝑥^∗‖^𝑝‖𝑦−𝑥‖^𝑝≤‖𝑦−𝑥‖^𝑝≤𝑟(𝑦)=𝑟(𝑦)−𝑟(𝑥)(𝑦∈𝑋). (2.2)It follows that sgn⟨𝑥^∗,𝑦−𝑥⟩|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝+𝑟(𝑥)≤𝑟(𝑦)(𝑦∈𝑋), and so 𝜕𝑟(𝑥)≠∅. Throughout the rest, we assume 𝜕𝑟(𝑥)≠∅ when we deal with this set. For an arbitrary nonempty bounded subset 𝑀 of 𝑋, finding the set of all 𝑥 for which 𝜕𝑟(𝑥)≠∅ remains an open question.

Lemma 2.1. Let 𝑋 be a 𝑝-Banach space and let 𝑀 be a bounded subset in 𝑋. Then for each 𝑥∈𝑋, each element of 𝜕𝑟(𝑥) has norm less than or equal to 1 and hence 𝜕𝑟(𝑥) is 𝑤^∗-compact.

Proof. Let 𝑥∈𝑋 and 𝑥^∗∈𝜕𝑟(𝑥). We have sgn⟨𝑥^∗,𝑦−𝑥⟩|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝+𝑟(𝑥)≤𝑟(𝑦)(𝑦∈𝑋). By definition of 𝑟(𝑥) we have |𝑟(𝑦)−𝑟(𝑥)|≤‖𝑥−𝑦‖^𝑝 for all 𝑦∈𝑋 [10, 12].
Hence sgn⟨𝑥^∗,𝑦−𝑥⟩|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝≤‖𝑥−𝑦‖^𝑝 and therefore ‖𝑥^∗‖≤1.

Note that −𝑟(𝑥)≤−‖𝑥−𝑦‖^𝑝≤−sgn⟨𝑥^∗,𝑥−𝑦⟩|⟨𝑥^∗,𝑥−𝑦⟩|^𝑝=sgn⟨𝑥^∗,𝑦−𝑥⟩|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝, thus inf_𝑧∈𝑀sgn⟨𝑥^∗,𝑧−𝑥⟩|⟨𝑥^∗,𝑧−𝑥⟩|^𝑝≥−𝑟(𝑥).

Now we have the following proposition which is interesting on its own right.

Proposition 2.2. Let 𝑋 be a 𝑝-Banach space and let 𝑀 be a bounded subset of 𝑋. Then the set 𝐹={𝑥∈𝑋∶inf_𝑧∈𝑀𝑠𝑔𝑛⟨𝑥^∗,𝑧−𝑥⟩|⟨𝑥^∗,𝑧−𝑥⟩|^𝑝>−𝑟(𝑥) for some 𝑥^∗∈𝜕𝑟(𝑥)} is of the first category in 𝑋.

Proof. Let𝐹_𝑛∶={𝑥∈𝑋∶inf_𝑧∈𝑀sgn⟨𝑥^∗,𝑧−𝑥⟩|⟨𝑥^∗,𝑧−𝑥⟩|^𝑝1≥−𝑟(𝑥)+𝑛forsome𝑥^∗∈𝜕𝑟(𝑥)}. (2.3) Then 𝐹=⋃^∞_𝑛=1𝐹_𝑛. We will show that for each 𝑛, (i)𝐹_𝑛 is a norm closed subset of 𝑋;(ii)𝐹_𝑛 has empty interior. To see (i), let {𝑥_𝑚} be a sequence in 𝐹_𝑛 which converges to an element 𝑥 in 𝑋. For each 𝑚, choose 𝑥^∗_𝑚∈𝜕𝑟(𝑥_𝑚) such thatinf_𝑧∈𝑀sgn⟨𝑥^∗_𝑚,𝑧−𝑥_𝑚⟩|⟨𝑥^∗_𝑚,𝑧−𝑥_𝑚⟩|^𝑝≥−𝑟(𝑥_𝑚1)+𝑛. (2.4)By Lemma 2.1‖𝑥^∗_𝑚‖≤1 for all 𝑚. Without loss of generality, we assume that {𝑥^∗_𝑚} converges weak^∗ to 𝑥^∗. For every 𝑦∈𝑋, we have|⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩−⟨𝑥^∗,𝑦−𝑥⟩|≤|⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩−⟨𝑥^∗_𝑚,𝑦−𝑥⟩|+|⟨𝑥^∗_𝑚,𝑦−𝑥⟩−⟨𝑥^∗,𝑦−𝑥⟩|≤‖𝑥_𝑚−𝑥‖+|⟨𝑥^∗_𝑚−𝑥^∗,𝑦−𝑥⟩|. (2.5)This shows that {⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩}^∞_𝑚=1 converges to ⟨𝑥^∗,𝑦−𝑥⟩. Since 𝑥^∗_𝑚∈𝜕𝑟(𝑥_𝑚),sgn⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|^𝑝+𝑟(𝑥_𝑚)≤𝑟(𝑦)(𝑦∈𝑋), (2.6)or equivalently⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|^𝑝−1+𝑟(𝑥_𝑚)≤𝑟(𝑦)(𝑦∈𝑋). (2.7)It follows that⟨𝑥^∗,𝑦−𝑥⟩|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝−1+𝑟(𝑥)≤𝑟(𝑦)(𝑦∈𝑋) (2.8)and hencesgn⟨𝑥^∗,𝑦−𝑥⟩|⟨𝑥^∗,𝑦−𝑥⟩|^𝑝+𝑟(𝑥)≤𝑟(𝑦)(𝑦∈𝑋). (2.9)This shows that 𝑥^∗∈𝜕𝑟(𝑥). It follows from (2.4) thatsgn⟨𝑥^∗_𝑚,𝑧−𝑥_𝑚⟩|⟨𝑥^∗_𝑚,𝑧−𝑥_𝑚⟩|^𝑝≥−𝑟(𝑥_𝑚1)+𝑛(𝑧∈𝑀). (2.10)We use the fact that sgn⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|^𝑝=⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|⟨𝑥^∗_𝑚,𝑦−𝑥_𝑚⟩|^𝑝−1 once more to obtain the inequalitysgn⟨𝑥^∗,𝑧−𝑥⟩|⟨𝑥^∗,𝑧−𝑥⟩|^𝑝1≥−𝑟(𝑥)+𝑛(𝑧∈𝑀). (2.11)Therefore 𝑥∈𝐹_𝑛. So 𝐹_𝑛 is a closed subset of 𝑋.
To see (ii), suppose that some 𝐹_𝑘 has nonempty interior. Then, there exists an open ball 𝑈 in 𝑋 of radius 𝜆(2𝑟(𝑦₀))^1/𝑝 for some 𝜆>0 and center at 𝑦₀ such that 𝑈⊆𝐹_𝑘. Let 𝛼=1/𝜆^𝑝,𝛽=1/(𝜆+1)^𝑝, and 𝜖≤((1+𝜆)^𝑝−1)/𝑘(𝛼+𝛽+1)((1+𝜆)^𝑝)min{𝑟^1/𝑝(𝑦₀),1} and choose 𝑧₀∈𝑀 such that𝑟(𝑦₀)−𝜖<‖𝑦₀−𝑧₀‖^𝑝≤𝑟(𝑦₀). (2.12)Let 𝑥₀=𝑦₀+𝜆(𝑦₀−𝑧₀) then𝑥₀−𝑧₀=(1+𝜆)(𝑦₀−𝑧₀). (2.13)Choose 𝑥₁∈𝑈⊆𝐹_𝑘 such that‖𝑥₁−𝑥₀‖^𝑝=𝜖. (2.14)Then there exists 𝑥^∗₁∈𝜕𝑟(𝑥₁) such thatinf_𝑧∈𝑀sgn⟨𝑥^∗₁,𝑧−𝑥₁⟩|⟨𝑥^∗₁,𝑧−𝑥₁⟩|^𝑝≥−𝑟(𝑥₁1)+𝑘. (2.15)We will show thatsgn⟨𝑥^∗₁,𝑦₀−𝑥₁⟩|⟨𝑥^∗₁,𝑦₀−𝑥₁⟩|^𝑝+𝑟(𝑥₁)>𝑟(𝑦₀). (2.16)This will contradict the fact that 𝑥^∗₁ is a subdifferential of 𝑟 at 𝑥₁ and the proof would be completed. To achieve a contradiction, we will consider four cases as follows. (i)sgn⟨𝑥^∗₁,𝑧₀−𝑥₁⟩<0 and sgn⟨𝑥^∗₁,𝑦₀−𝑥₁⟩>0.(ii)sgn⟨𝑥^∗₁,𝑧₀−𝑥₁⟩>0 and sgn⟨𝑥^∗₁,𝑦₀−𝑥₁⟩>0.(iii)sgn⟨𝑥^∗₁,𝑧₀−𝑥₁⟩>0 and sgn⟨𝑥^∗₁,𝑦₀−𝑥₁⟩<0.(iv)sgn⟨𝑥^∗₁,𝑧₀−𝑥₁⟩<0 and sgn⟨𝑥^∗₁,𝑦₀−𝑥₁⟩<0. We investigate case (i) in detail. The other cases can be studied similarly. First of all note that𝑧₀−𝑥₁=𝑧₀−𝑦₀+𝑦₀−𝑥₁=𝑦₀−𝑥₀𝜆+𝑦₀−𝑥₁1=(1+𝜆)(𝑦₀−𝑥₁1)+𝜆(𝑥₁−𝑥₀). (2.17)Now, we have𝑟(𝑦₀)−𝑟(𝑥₁)<‖𝑦₀−𝑧₀‖^𝑝+𝜖−𝑟(𝑥₁)=1|1+𝜆|^𝑝‖𝑥₀−𝑧₀‖^𝑝+𝜖−𝑟(𝑥₁)(by(2.13))≤1(1+𝜆)^𝑝[‖𝑥₀−𝑥₁‖^𝑝+‖𝑥₁−𝑧₀‖^𝑝]+𝜖−𝑟(𝑥₁)1=(1−(1+𝜆)^𝑝)(−𝑟(𝑥₁1))+((1+𝜆)^𝑝+1)𝜖(bythedefinationof1𝑟(⋅))≤(1−(1+𝜆)^𝑝)[sgn⟨𝑥^∗₁,𝑧₀−𝑥₁⟩|⟨𝑥^∗₁,𝑧₀−𝑥₁⟩|^𝑝−1𝑘1]+((1+𝜆)^𝑝1+1)𝜖≤(1−(1+𝜆)^𝑝)[−|⟨𝑥^∗₁,𝑧₀−𝑥₁⟩|^𝑝−1𝑘1]+((1+𝜆)^𝑝1+1)𝜖≤(1−(1+𝜆)^𝑝)[−|⟨𝑥^∗₁,𝜆+1𝜆(𝑦₀−𝑥₁𝑥)+¹−𝑥₀𝜆⟩|^𝑝−1𝑘]1+((1+𝜆)^𝑝+1)𝜖(by1(2.17))≤(1−(1+𝜆)^𝑝)[−|⟨𝑥^∗₁,𝜆+1𝜆(𝑦₀−𝑥₁)⟩|^𝑝+|⟨𝑥^∗₁,𝑥₁−𝑥₀𝜆⟩|^𝑝−1𝑘]1+((1+𝜆)^𝑝+1)𝜖(bythefactthat−|𝑎+𝑏|^𝑝≤−|𝑎|^𝑝+|𝑏|^𝑝)1≤(1−(1+𝜆)^𝑝)(𝜆+1𝜆)^𝑝[−⟨𝑥^∗₁,𝑦₀−𝑥₁⟩|^𝑝𝜖]+𝜆^𝑝1(1−(1+𝜆)^𝑝)1+(1−(1+𝜆)^𝑝1)(−𝑘1)+((1+𝜆)^𝑝1+1)𝜖≤(1−(1+𝜆)^𝑝)[−|⟨𝑥^∗₁,𝑦₀−𝑥₁⟩|^𝑝]≤sgn⟨𝑥^∗₁,𝑦₀−𝑥₁⟩|⟨𝑥^∗₁,𝑦₀−𝑥₁⟩|^𝑝. (2.18)

We recall that a set 𝐺⊂𝑋 is said to be weakly sequentially compact if each sequence of elements of 𝐺 contains a subsequence converging weakly to some element 𝑥∈𝐺.

Theorem 2.3. Let 𝑀 be a weakly sequentially compact subset in a 𝑝-Banach space 𝑋. Then the set {𝑥∈𝑋∶‖𝑥−𝑧‖^𝑝=𝑟(𝑥) for some 𝑧∈𝑀} contains a dense 𝐺_𝛿-set in 𝑋. In particular, the set of farthest points of 𝑆 is nonempty.

Proof. Let 𝐹 and 𝐹_𝑛 be defined as in Proposition 2.2 and let 𝐷(𝑀)=𝑋⧵𝐹. Then𝐷(𝑀)=𝑋⧵_𝑛∈𝑁𝐹_𝑛=_𝑛∈𝑁(𝑋⧵𝐹_𝑛), (2.19)where each 𝑋⧵𝐹_𝑛 is an open dense subset of 𝑋. Hence 𝐷(𝑀) is a dense 𝐺_𝛿-set in 𝑋. For each 𝑥∈𝐷(𝑀) and 𝑥^∗∈𝜕𝑟(𝑥), we haveinf_𝑧∈𝑀sgn⟨𝑥^∗,𝑧−𝑥⟩|⟨𝑥^∗,𝑧−𝑥⟩|^𝑝=−𝑟(𝑥). (2.20)By the weak compactness of 𝑀, there exists a point 𝑧∈𝑀 with sgn⟨𝑥^∗,𝑧₀−𝑥⟩|⟨𝑥^∗,𝑧₀−𝑥⟩|^𝑝=−𝑟(𝑥). Hence𝑟(𝑥)≥‖𝑥−𝑧₀‖^𝑝≥sgn⟨𝑥^∗,𝑥−𝑧₀⟩|⟨𝑥^∗,𝑥−𝑧₀⟩|^𝑝=−sgn⟨𝑥^∗,𝑧₀−𝑥⟩|⟨𝑥^∗,𝑧₀−𝑥⟩|^𝑝=𝑟(𝑥). (2.21)This shows that 𝐷(𝑀)⊆{𝑥∶‖𝑥−𝑧‖^𝑝=𝑟(𝑥) for some 𝑧∈𝑀}.

Acknowledgment

The authors would like to thank the referee for valuable comments.