# A Characterization for Compact Sets in the Space of Fuzzy Star-Shaped Numbers withLp Metric

Authors: Zhitao Zhao; Congxin Wu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By means of some auxiliary lemmas, we obtain a characterization of compact
subsets in the space of all fuzzy star-shaped numbers withLp metric for 1≤p<∞. The result further completes and develops the previous characterization of compact subsets given by Wu and Zhao in 2008.


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## Body

## 1. Introduction

Since the concept of fuzzy numbers was firstly introduced in 1970s, it has been studied extensively from many different aspects of the theory and applications such as fuzzy algebra, fuzzy analysis, fuzzy topology, fuzzy logic, and fuzzy decision making. Many applications restrict their description to fuzzy numbers, often implicitly, because of powerful fuzzy convexity of fuzzy numbers. This fuzzy convexity is mainly reflected on the convexity of the level sets of fuzzy numbers. However, apart from possible applications, it is of independent interest to see how far the supposition of convexity can be weakened without losing too much structure. Star-shapedness is a fairly natural extension to convexity. Surprisingly, many topological properties of spaces of compact star-shaped sets are similar to those of their compact convex counterparts. It is well known that star-shapedness also plays an important role in the theory and applications, such as nonsmooth analysis, approximation problems and optimization problems, (see [1–11]). Based on the importance of star-shapedness, as a corresponding extension to fuzzy numbers, fuzzy star-shaped numbers have been payed more and more attention, such as Chanussot et al. [12], Diamond [13], Diamond and Kloeden [14], Qiu et al. [15], and Wu and Zhao [16].In this regard, Diamond has done a lot of work. In 1990, he firstly introduced the concept of fuzzy star-shaped numbers and defined metrics on the space of fuzzy star-shaped numbers in [13]. Especially, he studied the properties of Lp metric for 1≤p<∞. The induced metric spaces (S0n,dp) were shown to be separable, but not complete for 1≤p<∞. Finally, compact sets in (S0n,dp) were also characterized. Later on, Wu and Zhao [16] pointed out that the characterization of Diamond was incorrect by a counterexample and gave a correct characterization for compact sets in spaces (S0n,dp) for 1≤p<∞.The aim of this paper is to further improve the characterization of compact sets in [16] and obtain a characterization of compact sets in the space of all fuzzy star-shaped numbers with Lp metric for 1≤p<∞. The result of this paper will provide some help for future research on fuzzy star-shaped numbers.


## 2. Preliminaries

Inℝn, denote the Euclidean norm by ∥·∥, and denote the class of all nonempty compact sets in ℝn by 𝕂n. If A,B are nonempty compact sets in ℝn, then the Hausdorff distance between A and B is given by dH(A,B)=max{supa∈Ainfb∈B∥a-b∥,supb∈Binfa∈A∥a-b∥}.Definition 1 (see [14]).
An elementK∈𝕂n is star-shaped relative to a point x∈K if for each y∈K, the line segment xy¯ joining x to y is contained in K. The kerK of K is the set of all points x∈K such that the line segment xy¯⊂K for each y∈K and coK is the convex hull of K.For a fuzzy setu:ℝn→[0,1], we suppose that(1)u is normal; that is, there exists an x0∈ℝn such that u(x0)=1,(2)u is upper semicontinuous,(3)suppu=cl{x∈ℝn:u(x)>0} is compact,(4)u is fuzzy star-shaped; that is, there exists x∈ℝn such that u is fuzzy star-shaped with respect to x, namely, for any y∈ℝn and 0≤λ≤1,
(1)u(λy+(1-λ)x)≥u(y),(4°)u is fuzzy star-shaped with respect to the origin.Definition 2 (see [16]).
A fuzzy star-shaped number is a fuzzy setu:ℝn→[0,1] satisfying (1), (2), (3), and (4). Let Sn be the family of all fuzzy star-shaped numbers, and let S0n be the family of all fuzzy sets which satisfy (1), (2), (3), and (4°). Clearly, S0n⊂Sn.For a fuzzy star-shaped numberu∈Sn, we define its α-level set as follows:
(2)[u]α={{x∈ℝn:u(x)≥α},if0<α≤1,suppu,ifα=0.Lettingm∈ℝn, we define
(3)m^(x)={1,ifx=m,0,ifx≠m.
It is obvious that m^∈Sn for each m∈ℝn.We will define addition and scalar multiplication of fuzzy shaped-numbers levelsetwise; that is, foru,v∈Sn and c≥0(4)[u+v]α=[u]α+[v]α,[cu]α=c[u]α,
for each α∈[0,1]. It directly follows that u+v,cu∈Sn.Now, let us recall some properties of fuzzy star-shaped numbers which will be used in this paper.Proposition 3.
Ifu∈Sn, then(1)
[u]β⊆[u]α⊆[u]0, 0≤α≤β≤1,(2)
[u]α is a compact set of ℝn for each 0≤α≤1,(3)
for anyα∈(0,1], if {αk} is an increasing sequence of real numbers in (0,1] converging to α, then [u]α=⋂k=1∞[u]αk.The following property derives directly from Definitions1 and 2.Proposition 4 (see [14]).
For anyu∈Sn, u is fuzzy star-shaped with respect to y if and only if [u]α are star-shaped with respect to y for all α∈[0,1].Definition 5 (see [14]).
Letker(u) be the totality of y∈ℝn such that u is fuzzy star-shaped with respect to y. Define fker(u) by [fker(u)]α=ker[u]α for a fuzzy star-shaped number u.Proposition 6 (see [14]).
For a fuzzy star-shaped numberu, ker(u) is a convex set in ℝn, and fker(u) is a fuzzy convex set which is normal; that is, [fker(u)]1≠∅.Definition 7 (see [14]).
For each1≤p<∞, one defines
(5)dp(u,v)=(∫01‍dH([u]α,[v]α)pdα)1/p,
for all u,v∈Sn, then dp is called the Lp metric on Sn.For the characterization of compact sets in (Sn,dp), the following definitions will be used in the sequel.Definition 8 (see [14]).
One says thatU⊂Sn is uniformly support bounded if the support sets [u]0 are bounded in ℝn, uniformly for u∈U; that is, there is a constant 𝒦>0 such that dH([u]0,{θ})≤𝒦 holds for all u∈U.Definition 9 (see [14]).
Letu∈Sn. If for any ε>0, there exists δ=δ(u,ε)>0 such that for all 0≤h<δ(6)∫h1‍dH([u]α,[u]α-h)pdα<εp,1≤p<∞,
then one says that u is p-mean left-continuous. If for nonempty U⊂Sn, the above inequality holds uniformly for u∈U, one says that U is p-mean equi-left-continuous.Definition 10 (see [17]).
One says thatU⊂Sn is uniformly p-mean bounded if there is a constant 𝒦>0 such that dp(u,θ^)≤𝒦 for all u∈U, where θ^ denotes u∈Sn with u(θ)=1, and u(x)=0 for any x≠θ.Remark 11.
Note that uniformp-mean boundedness is weaker than uniform support boundedness, namely, if U⊂(Sn,dp) is uniformly support bounded, then U is uniformly p-mean bounded; however, the converse implication does not hold.


## 3. Main Results

Before proving our main result, we demonstrate some auxiliary lemmas.Lemma 12.
Letu∈Sn, then the following properties hold:(1)
ker(u)⊂[u]1,(2)
for anym∈ker(u), u-m^ exists, and for any α∈[0,1], one has
(7)[u-m^]α=[u]α-{m},[u+m^]α=[u]α+{m},(3)
for anym∈ker(u), u-m^∈S0n,(4)
if{mk}⊂ℝn converges to m0∈ℝn, then {m^k} converges to m^0 in (Sn,dp).Proof.
(1) Letm∈ker(u). By Definition 5, we know that for all x∈ℝn(8)u(λx+(1-λ)m)≥u(x),0≤λ≤1.
Especially, the above inequality holds for each x∈[u]1 and λ=0; that is, u(m)≥u(x)=1. Thus, we get m∈[u]1. Since m is an arbitrary element of ker(u), then we have that ker(u)⊂[u]1.
(2) According to the Zadeh extension principle, we have(9)(u-m^)(z)=supz=x-ymin{u(x),m^(y)},
and so, it is clear that u-m^ exists for each m∈ker(u). In the following, we infer that [u-m^]α=[u]α-{m} for each α∈[0,1].
Letα∈[0,1]. For any z∈[u]α-{m}, there exists x0∈[u]α such that z=x0-m. Then
(10)(u-m^)(z)=supz=x-ymin{u(x),m^(y)}≥min{u(x0),m^(m)}=u(x0)≥α,
and so, z∈[u-m^]α. On the other hand, for any z∈[u-m^]α, by the Zadeh extension principle, we get
(11)(u-m^)(z)=supz=x-ymin{u(x),m^(y)}=supz=x-mu(x)≥α.
Then, by the definition of the supremum, there exists xk∈[u]0 for each k∈ℕ such that
(12)u(xk)≥α-1k,z=xk-m.
From the compactness of[u]0, {xk} has a subsequence {xkj} converging to x0∈[u]0. Since u is upper semicontinuous, then we have
(13)u(x0)≥lim¯j→∞u(xkj)≥α.
That is, x0∈[u]α; thus, z∈[u]α-{m}.
Applying the similar techniques, we obtain[u+m^]α=[u]α+{m} for each α∈[0,1].
(3) By Proposition4, to prove u-m^∈S0n, it is enough to verify that [u-m^]α is star-shaped with respect to the origin for each α∈[0,1], where m is an arbitrary element of ker(u).
For anyx∈[u-m^]α, α∈[0,1]; by statement (2), we have that x∈[u]α-{m}. So, we obtain that x+m∈[u]α. Since m is an arbitrary element of ker(u), then by Definition 5 and Proposition 4, the line segment m(x+m)¯ joining m to x+m is contained in [u]α; that is,
(14)λ(x+m)+(1-λ)m=λx+m∈[u]α,0≤λ≤1.
Thus, λx∈[u]α-{m} for all 0≤λ≤1. That is, the line segment θx¯⊂[u]α-{m}.
Consequently,[u-m^]α is star-shaped with respect to the origin for each α∈[0,1].
(4) Since{mk}⊂ℝn converges to m0∈ℝn, then for every ε>0, there exists an integer k0 such that when k≥k0(15)∥mk-m0∥<ε.
Thus, for the aboveε>0, we have
(16)dp(m^k,m^0)=(∫01‍dH([m^k]α,[m^0]α)pdα)1/p=(∫01‍dH({mk},{m0})pdα)1/p=(∫01‍∥mk-m0∥pdα)1/p<ε,
whenever k≥k0. Therefore, {m^k} converges to m^0 in (Sn,dp).Lemma 13.
For anyu∈(Sn,dp), 1≤p<∞, u is p-mean left-continuous.Proof.
The technique is similar to the proof of Lemma 3.1 in [16].For everyu∈Sn and r∈[0,1], denote
(17)u(r)(x)={u(x),ifu(x)≥r,0,ifu(x)<r.
Then, we have the following.Lemma 14 (see [16]).
A closed setU⊂(S0n,dp), 1≤p<∞, is compact if and only if(1)
U is uniformly p-mean bounded,(2)
U is p-mean equi-left-continuous,(3)
let{ri} be a decreasing sequence in (0,1] converging to zero. For {uk}⊂U, if {uk(ri)∣k=1,2,3,…} converges to u(ri)∈S0n in dp, then there exists a u0∈S0n such that
(18)[u0(ri)]α=[u(ri)]α,ri<α≤1.Lemma 15.
For anyu∈Sn,m∈ℝn, and 0<r<1, one has
(19)m^(r)=m^,(u+m^)(r)=u(r)+m^.Proof.
Letα be an arbitrary element in [0,1].
Ifr≤α≤1, then [m^(r)]α=[m^]α={m}. If 0<α<r, then [m^(r)]α=[m^]r={m}=[m^]α. Hence, for each α∈(0,1],[m^(r)]α=[m^]α, and so, it follows that [m^(r)]0=[m^]0. This implies that m^(r)=m^.
Similarly, ifr≤α≤1, then by Lemma 12(2), we have
(20)[(u+m^)(r)]α=[u+m^]α=[u]α+{m}=[u(r)]α+{m}=[u(r)+m^]α.
If 0<α<r, then by Lemma 12(2), we have
(21)[(u+m^)(r)]α=[u+m^]r=[u]r+{m}=[u(r)]α+{m}=[u(r)+m^]α.
Hence, for eachα∈(0,1],[(u+m^)(r)]α=[u(r)+m^]α, and so, it follows that [(u+m^)(r)]0=[u(r)+m^]0. This implies that (u+m^)(r)=u(r)+m^.Now, we give the characterization of compact sets in(Sn,dp) for 1≤p<∞.Theorem 16.
A closed setU⊂(Sn,dp), 1≤p<∞ is compact if and only if(i)
U is uniformly p-mean bounded,(ii)
U is p-mean equi-left-continuous,(iii)
let{ri} be a decreasing sequence in (0,1] converging to zero. For {uk}⊂U, if {uk(ri)∣k=1,2,3,…} converges to u(ri)∈Sn in dp, then there exists a u0∈Sn such that
(22)[u0(ri)]α=[u(ri)]α,ri<α≤1.Proof.
Necessity: (1) Since U is compact in (Sn,dp), it follows that U is a bounded set in (Sn,dp). This implies that U is uniformly p-mean bounded.
(2) Letε>0, and let u1,u2,…,um∈Sn be a (ε/3)-net of U, that is, for any u∈U, there exists an element ui(1≤i≤m) satisfying dp(u,ui)<ε/3. By Lemma 12, u1,u2,…,um are p-mean left-continuous, and so, there exists δ(ε)=min1≤i≤mδ(ui,ε)>0 such that
(23)∫h1‍dH([ui]α,[ui]α-h)pdα<(ε3)p,
for i=1,2,…,m and 0≤h<δ(ε).
Thus, foru∈U, we can obtain by triangle inequality
(24)(∫h1‍dH([u]α,[u]α-h)pdα)1/p≤(∫h1‍dH([u]α,[ui]α)pdα)1/p+(∫h1‍dH([ui]α,[ui]α-h)pdα)1/p+(∫h1‍dH([ui]α-h,[u]α-h)pdα)1/p≤dp(ui,u)+(∫h1‍dH([ui]α,[ui]α-h)pdα)1/p+dp(ui,u)<ε3+ε3+ε3=ε.
Therefore, U is p-mean equi-left-continuous.
(3) According to the definition ofu(ri), [u(ri)]α=[u]α for each ri≤α≤1, and [u(ri)]α=[u]ri for each 0≤α<ri. Since u∈Sn, then from Proposition 4, there exists x∈ℝn such that [u]α is star-shaped with respect to x for each 0≤α≤1. So, [u(ri)]α is star-shaped with respect to x for each 0≤α≤1; thus, it is obvious by Proposition 4 that u(ri)∈Sn for each i∈ℕ.
For{uk}⊂U, we assume that {uk(ri)}k=1∞ converges to u(ri)∈Sn in dp. Since U is compact in (Sn,dp), then {uk} has a subsequence {ukj} converging to u0∈Sn in dp. By Minkowski’s inequality, it follows that
(25)(∫ri1‍dH([u0]α,[u(ri)]α)pdα)1/p≤(∫ri1‍dH([u0]α,[ukj]α)pdα)1/p+(∫ri1‍dH([ukj]α,[u(ri)]α)pdα)1/p=(∫ri1‍dH([u0]α,[ukj]α)pdα)1/p+(∫ri1‍dH([ukj(ri)]α,[u(ri)]α)pdα)1/p.
But since
(26)limj→∞(∫ri1‍dH([u0]α,[ukj]α)pdα)1/p=0,limj→∞(∫ri1‍dH([ukj(ri)]α,[u(ri)]α)pdα)1/p=0,
then we get
(27)(∫ri1‍dH([u0]α,[u(ri)]α)pdα)1/p=0,
and this implies that
(28)dH([u0]α,[u(ri)]α)=0,
a.e. in (ri,1]. Thus, we obtain that
(29)[u0]α=[u(ri)]α,
a.e. in (ri,1]; that is, [u0]α=[u(ri)]α for each (ri,1]∖Ω, where Ω⊂(ri,1] has Lebesgue measure zero and clearly (ri,1]∖Ω is dense in (ri,1]. Let α-∈Ω, then there exists an increasing sequence αk↗α- with αk∈(ri,1]∖Ω and [u0]αk=[u(ri)]αk. So
(30)dH([u0]α-,[u(ri)]α-)≤dH([u0]α-,[u0]αk)+dH([u0]αk,[u(ri)]αk)+dH([u(ri)]αk,[u(ri)]α-)=dH([u0]α-,[u0]αk)+dH([u(ri)]αk,[u(ri)]α-).
By Proposition 3, both terms on the right converge to zero as k→∞, so dH([u0]α-,[u(ri)]α-)=0, and hence [u0]α-=[u(ri)]α- for each α-∈Ω.
Consequently, we have[u0(ri)]α=[u0]α=[u(ri)]α for each ri<α≤1.
Sufficiency: From Lemma 12(3), we have that {u-m^:u∈U,m∈ker(u)}⊂S0n. We divide the rest of the proof into two steps.
Step  1. We prove that {u-m^:u∈U,m∈ker(u)} satisfies (1)–(3) in Lemma 14.
Firstly, we show that ifU is uniformly p-mean bounded, then {ker(u):u∈U} is uniformly bounded. Otherwise, we infer that for any k>0, k∈ℕ; we can find that uk∈U such that
(31)dH(ker(uk),{θ})>k.
By Proposition3(1) and Lemma 12(1), it follows that
(32)(∫01‍dH([uk]α,{θ})pdα)1/p≥(∫01‍dH([uk]1,{θ})pdα)1/p≥(∫01‍dH(ker(uk),{θ})pdα)1/p>(∫01‍kpdα)1/p=k,k∈ℕ.
This contradicts condition (i).
Secondly, we conclude that{u-m^:u∈U,m∈ker(u)} is uniformly p-mean bounded. According to condition (i) and the conclusion in last paragraph, we obtain that {ker(u):u∈U} is uniformly bounded, and then there exists a 𝒦>0 such that for all u∈U(33)dH(ker(u),{θ})≤𝒦.
By the condition (i), there exists a 𝒦′>0 such that for all u∈U(34)(∫01‍dH([u]α,{θ})pdα)1/p≤𝒦′.
Thus, by Lemma12(2) for all u∈U, we get
(35)(∫01‍dH([u-m^]α,{θ})pdα)1/p=(∫01‍dH([u]α-{m},{θ})pdα)1/p=(∫01‍dH([u]α,{m})pdα)1/p≤(∫01‍dH([u]α,{θ})pdα)1/p+(∫01‍dH({θ},{m})pdα)1/p≤(∫01‍dH([u]α,{θ})pdα)1/p+(∫01‍dH({θ},ker(u))pdα)1/p≤𝒦′+𝒦.
This implies that condition (1) of Lemma 14 is satisfied.
Thirdly, we verify that{u-m^:u∈U,m∈ker(u)} is p-mean equi-left-continuous. In fact, since U is p-mean equi-left-continuous, then for every ε>0, there exists a δ>0 such that for all 0≤h≤δ and u∈U(36)∫h1‍dH([u]α,[u]α-h)pdα<εp,
and so, we have
(37)∫h1‍dH([u-m^]α,[u-m^]α-h)pdα=∫h1‍dH([u]α-{m},[u]α-h-{m})pdα=∫h1‍dH([u]α,[u]α-h)pdα<εp.
Hence, condition (2) of Lemma 14 holds.
Finally, we prove that condition (3) of Lemma14 is also satisfied. Let {ri} be a decreasing sequence in (0,1] converging to zero. We suppose that {(uk-m^k)(ri)}(uk∈U,mk∈ker(uk)) converges to u*(ri)∈S0n in dp. Denote vk=uk-m^k. Then, {vk(ri)} converges to u*(ri)∈S0n in dp. By Lemma 15, we have
(38)uk(ri)=(vk+m^k)(ri)=vk(ri)+m^k.
By the preceding proof,{ker(u)∣u∈U} is uniformly bounded. Then, {mk} has a subsequence {mkj} converging to m0∈ℝn, and so by Lemma 12(4), {ukj(ri)}={vkj(ri)+m^kj} converges to u*(ri)+m^0 in (Sn,dp). By condition (iii) and Lemma 12(2), there exists a u0∈Sn such that
(39)[u0(ri)]α=[u*(ri)+m^0]α=[u*(ri)]α+{m0},
wherever ri<α≤1, and so by Lemma 12(2), we have
(40)[u*(ri)]α=[u0(ri)]α-{m0}=[u0]α-{m0}=[u0-m^0]α=[(u0-m^0)(ri)]α,
wherever ri<α≤1.
Defineu0*=u0-m^0. Then, by Lemma 12(3), it follows that u0*∈S0n. Thus, condition (3) of Lemma 14 holds.
Step  2. We infer that U is a compact set in (Sn,dp).
Let{uk}⊂U. Since {u-m^:u∈U,m∈ker(u)} is a subset of (S0n,dp) and satisfies condition (1)–(3) of Lemma 14, then {u-m^:u∈U,m∈ker(u)} is a compact set of (S0n,dp), and so for fixed mk∈ker(uk)  (k∈ℕ), {uk-m^k} has a subsequence {ukj-m^kj} converging to u0∈(S0n,dp). By the proceeding proof, {ker(u):u∈U} is uniformly bounded, so {mkj} has a subsequence {mkj(l)} converging to m0∈ℝn. It is obvious that {ukj(l)-m^kj(l)} converges to u0∈(S0n,dp).
Denotevkj(l)=ukj(l)-m^kj(l). Then, {vkj(l)} converges to u0∈(S0n,dp). Therefore, by Lemma 12(4), we get that {ukj(l)}={vkj(l)+m^kj(l)} converges to u0+m^0∈(Sn,dp), which completes the proof.


## 4. Conclusion

Since star-shapedness has played an important role in the theory and applications of classical analysis, such as nonsmooth analysis, approximation problems, and optimization problems, then as an extension to fuzzy numbers, fuzzy star-shaped numbers should be also payed more and more attention. In this paper, we further complete and develop the previous result in [16] and give a characterization of compact subsets in the space of all fuzzy star-shaped numbers with Lp metric for 1≤p<∞. The result of this paper will provide some help for future research on the theory of fuzzy star-shaped numbers.



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