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1 This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

2 Available online at Fuzzy Sets and Systems 2 (212) Lipschitz functions and fuzzy number approximations Lucian Coroianu a,b, a Department of Mathematics and Informatics, University of Oradea, Universităţii 1, 4187 Oradea, Romania b Department of Mathematics, Babes-Bolyai University of Cluj-Napoca, Mihail Kogalniceanu 1, 484 Cluj-Napoca, Romania Received September 211; received in revised form 1 December 211; accepted 2 January 212 Available online 2 January 212 Abstract We prove that some important properties of convex subsets of vector topological spaces remain valid in the space of fuzzy numbers endowed with the Euclidean distance. We use these results to obtain a characterization of fuzzy number-valued Lipschitz functions. This fact helps us to find the best Lipschitz constant of the trapezoidal approximation operator preserving the value and ambiguity introduced in a recent paper. Finally, applications in finding within a reasonable error the trapezoidal approximation of a fuzzy number preserving the value and ambiguity in the case when the direct formula is not applicable and an estimation for the defect of additivity of the trapezoidal approximation preserving the value and ambiguity are given. 212 Elsevier B.V. All rights reserved. Keywords: Fuzzy number; Trapezoidal fuzzy number; Approximation; Ambiguity; Value; Convexity; Lipschitz constant 1. Introduction In the last few years many papers investigated on the approximation of fuzzy numbers with respect to well-known metrics [1 5,7 1,15,16,21 26,28,2 8]. Mostly, two kinds of problems are considered: approximations without any other restriction and approximations with additional conditions. We recall here some important contributions with respect to the problem of the approximation of fuzzy numbers. Firstly, we discuss about approximations without additional conditions. Chanas [15] and Grzegorzewski [21] independently proposed the interval approximation of a fuzzy number. Grzegorzewski proved that the nearest interval approximation of a fuzzy number with respect to the Euclidean metric is actually its expected interval. Abbasbandy and Asady [] proposed the trapezoidal approximation of a fuzzy number with respect to the same Euclidean distance. Yeh [4] proposed new algorithms for computing trapezoidal and triangular approximations of fuzzy numbers. Zeng and Li [8] proposed the triangular approximation of a fuzzy number with respect to a weighted L 2 -type metric. Unfortunately, the algorithm proposed by them does not always produce proper triangular fuzzy numbers as it was pointed out in the papers [9,4]. The correct algorithm is given by Ban in the paper [9]. The most general result in approximations with trapezoidal or triangular fuzzy numbers is given by Yeh in the paper [7] where algorithms for computing trapezoidal or triangular approximations of fuzzy numbers Correspondence address: Department of Mathematics and Informatics, University of Oradea, Universităţii 1, 4187 Oradea, Romania. Fax: address: lcoroianu@uoradea.ro /$ - see front matter 212 Elsevier B.V. All rights reserved. doi:1.116/j.fss

3 L. Coroianu / Fuzzy Sets and Systems 2 (212) with respect to general weighted L 2 -type metrics are given. A generalization of the notion of trapezoidal fuzzy number is the parametric (also known as semi-trapezoidal) fuzzy number introduced by Nasibov and Peker in [28]. In the same paper they proposed the parametric approximation of a fuzzy number with respect to the Euclidean metric, result improved by Ban in [8]. ThenYeh[6] generalized these results by considering general weighted L 2 -type metrics. Now, we discuss about approximations of fuzzy numbers under additional conditions. Grzegorzewski and Mrówka [22,2] proposed the trapezoidal approximation of a fuzzy number preserving the expected interval with respect to the Euclidean metric. Then Ban [7] and Yeh [5] independently improved their result. Algorithms for computing the trapezoidal approximation of a fuzzy number preserving the expected interval can be found in the papers [24,25].Then Abbasbandy and Hajjari [4] proposed approximations of fuzzy numbers preserving the core with respect to weighted L 2 -type metrics. Recently, Ban and all [1] proposed the trapezoidal approximation preserving the value and ambiguity with respect to the Euclidean metric. Interestingly, the problem of approximating fuzzy numbers by trapezoidal fuzzy numbers preserving the value and ambiguity goes back to a paper of Delgado and all [18] where they considered that two fuzzy numbers with the same value and ambiguity should be considered equal. More generally, Ban and Coroianu [1] proposed simpler methods to compute the parametric approximation of a fuzzy number with respect to the Euclidean metric preserving important characteristics such as expected interval or the ambiguity and value. Other types of nonlinear approximations can be found in the paper of Grzegorzewski and Stefanini [26] where they proposed classes of fuzzy numbers depending on four or five parameters which allow approximations with conservation of multiple characteristics of fuzzy numbers such as the support and core or the ambiguity and value. The quality of a trapezoidal, triangular or parametric approximation operator is important nevertheless. For this reason, Grzegorzewski and Mrówka [22] proposed a list of criteria that a trapezoidal approximation operator should posses. Most of these approximation operators own important properties such as: translation invariance, scale invariance, or identity criteria. Another important property that an approximation operator should posses is the continuity. One would expect that if fuzzy number A is close to fuzzy number B then their approximations are also close one to another. Yeh [4,6,7] proved that the approximation operators without additional conditions are nonexpansives. Ban and Coroianu [11] proved that the trapezoidal approximation operator preserving the expected interval satisfies the Lipschitz condition. Then Coroianu [16] found the best Lipschitz constant of the discussed operator. Recently, in the paper [1] it was proved that the trapezoidal approximation operator preserving the value and ambiguity satisfies the Lipschitz condition too. As a negative result, in the paper [12] it was proved that any trapezoidal fuzzy number-valued operator (not necessarily an approximation operator) preserving core is discontinuous with respect to any weighted L 2 -type metric and each fuzzy number with the 1-cut set as a proper interval is a point of discontinuity. The aim of this paper is to provide a method which simplifies the investigation on distance properties of the trapezoidal approximation operators and especially on the finding of the best Lipschitz constant if possible. This is motivated by the fact that the method used in the case of the trapezoidal approximation preserving the expected interval (see the proof of Theorem 7 in [16]) cannot be used in general. Indeed, while in the case of the trapezoidal approximation preserving the expected interval the problem is reduced to geometrical reasonings in the Euclidean space R 2 (this is possible because two out of the four components of the trapezoidal approximation are given by formulas which allow best possible estimations), in general all the four components of the trapezoidal approximation are needed to obtain better estimations. This seems to be the case of the trapezoidal approximation preserving the value and ambiguity where the method proposed in the case of the trapezoidal approximation operator preserving the expected interval seems to be much more difficult to apply. However, most of the approximation operators including the two operators mentioned before have something in common. They are given on cases each one corresponding to a closed convex subset of the space of fuzzy numbers. When we say convex subset of the space of fuzzy numbers we are aware of the fact that the concept of convexity is usually in relation with the notion of vector space. Still, even if the addition and the scalar multiplication of fuzzy numbers have not the necessary properties to provide a vector structure over the space of fuzzy numbers, we can define the notion of convex subset of the space of fuzzy numbers exactly as in the case of vector spaces. The idea is not new, Prakash and Sertel introduced the notion of convex subset in semivector topological spaces in [29]. Actually, one can easily check that the space of fuzzy numbers is a semivector (nowadays called semilinear) space according to Prakash and Sertel definition, with respect to the addition and scalar multiplication of fuzzy numbers using their α-cuts. Moreover, we will prove that most of the elementary properties satisfied by some convex subsets of normed vector spaces remain valid in the case of the space of fuzzy numbers endowed with the Euclidean metric and we refer here to the results from Sections and 4. These properties will allow us to obtain a characterization of Lipschitz functions (Theorem 9 in Section 4) which is very suitable with the form of the trapezoidal approximation operators

4 118 L. Coroianu / Fuzzy Sets and Systems 2 (212) given on cases as we have mentioned before. From this characterization, as a direct application we will obtain the best Lipschitz constant of the trapezoidal approximation preserving the value and ambiguity (Theorem 1 in Section 5). Theorem 9 is quite general in the sense that it can be applied for most of the approximation operators known in the literature since most of them fulfil the requirements of Theorem 9. Finally, in Section 6 we propose two applications. Firstly, we will find within a reasonable error the trapezoidal approximation of a fuzzy number in the case when the direct formula is not applicable. Then we find an estimation for the defect of additivity of the discussed operator which is an improvement of the result proposed in the paper [1]. 2. Preliminaries We consider the following well-known description of a fuzzy number A: if x a 1, l A (x) if a 1 x a 2, A(x) = 1 if a 2 x a, r A (x) if a x a 4, if a 4 x, (1) where a 1, a 2, a, a 4 R, l A :[a 1, a 2 ] [, 1] is a nondecreasing upper semicontinuous function, l A (a 1 ) =, l A (a 2 ) = 1, called the left side of the fuzzy number and r A :[a, a 4 ] [, 1] is a nonincreasing upper semicontinuous function, r A (a ) = 1, r A (a 4 ) =, called the right side of the fuzzy number. The α-cut, α (, 1], of a fuzzy number A is a crisp set defined as A α ={x R : A(x) α}. The support or -cut A of a fuzzy number is defined as A = {x R : A(x) > }. Every α-cut α [, 1], of a fuzzy number is a closed interval where A α = [A L (α), A U (α)], A L (α) = inf{x R : A(x) α}, A U (α) = sup{x R : A(x) α} for any α (, 1]. If the sides of the fuzzy number A are strictly monotone then one can see easily that A L and A U are inverse functions of l A and r A, respectively. We denote by F(R) the set of all fuzzy numbers. In this paper we say that the fuzzy numbers A and B are equal and we denote A=B if A L (α) = B L (α), A U (α) = B U (α) almost everywhere α [, 1]. Some important notions connected with the concept of fuzzy number are the ambiguity Amb(A) and the value Val(A) of a fuzzy number A, A α = [A L (α), A U (α)], α [, 1]. They are given by (see [18]) Amb(A) = Val(A) = α(a U (α) A L (α)) dα, α(a U (α) + A L (α)) dα. (2) ()

5 L. Coroianu / Fuzzy Sets and Systems 2 (212) In practical problems such as the solving of fuzzy equations, data analysis or the ranking of fuzzy numbers, it is extremely important to endow the space of fuzzy numbers with topological structures. The flexibility of the space of fuzzy numbers allows the construction of many types of metric structures over this space. For example regarding the space of fuzzy numbers as a space of functions, the classical Chebyshev type distance between functions can be defined (see e.g. [17, p. 51]). Perhaps one of the most important topological structures over the space of fuzzy numbers is the one obtained from the Hausdorff metric between compact subsets of R 2, introduced by Goetschel and Voxman in [19] (see also [17, Chapter 7] and the references given at the end of the chapter). Usually a metric is defined with a precise purpose. For instance, in the paper [27] different metrics between fuzzy numbers are introduced in order to rank fuzzy numbers. Then, in the papers [14,1] new metrics are proposed with important advantages in statistical problems. The present paper deals with the problem of the approximation of fuzzy numbers. It seems that the suitable metric for this type of problem is an extension of the Euclidean distance and it is defined by [2] d 2 (A, B) = (A L (α) B L (α)) 2 dα + (A U (α) B U (α)) 2 dα. (4) Generalizations of the above L 2 -type metric can be found in the papers [7,8] as they are usually called weighted L 2 -type metrics. In all that follows in this paper, we will deal only with the case of the above Euclidean metric d since the main application of the paper is in relation with this particular metric. Fuzzy numbers with simple membership functions are preferred in practice. The most used such fuzzy numbers are so-called trapezoidal fuzzy numbers given by if x t 1, x t 1 if t 1 x t 2, t 2 t 1 T (x) = 1 if t 2 x t, t 4 x if t x t 4, t 4 t if t 4 x. When t 2 = t we obtain a so-called triangular fuzzy number. We denote T = (t 1, t 2, t, t 4 ) a trapezoidal fuzzy number as above. It is easy to prove that and T L (α) = t 1 + (t 2 t 1 )α T U (α) = t 4 (t 4 t )α for every α [, 1]. If T = (t 1, t 2, t, t 4 )andt = (t 1, t 2, t, t 4 ) then applying formula (4) weget d 2 (T, T ) = 1 (t 1 t 1 )2 + 1 (t 2 t 2 )2 + 1 (t 1 t 1 )(t 2 t 2 ) + 1 (t t )2 + 1 (t 4 t 4 )2 + 1 (t t )(t 4 t 4 ). (5) We denote by F T (R) the set of all trapezoidal fuzzy numbers. Let A, B F(R), A α = [A L (α), A U (α)], B α = [B L (α), B U (α)], α [, 1] and λ R. We consider the sum A+B and the scalar multiplication λ A by (see e.g. [17, p. 4]) (A + B) α = A α + B α = [A L (α) + B L (α), A U (α) + B U (α)] and { [λal (α), λa U (α)] if λ, (λ A) α = λa α = [λa U (α), λa L (α)] if λ <,

6 12 L. Coroianu / Fuzzy Sets and Systems 2 (212) respectively, for every α [, 1]. In the case of the trapezoidal fuzzy numbers T = (t 1, t 2, t, t 4 )ands = (s 1, s 2, s, s 4 ) we obtain T + S = (t 1 + s 1, t 2 + s 2, t + s, t 4 + s 4 ). If (X 1, d 1 )and(x 2, d 2 ) are metric spaces then a function f : X 1 X 2 is called a Lipschitz function if there exists a real positive constant C such that d 2 ( f (x 1 ), f (x 2 )) Cd 1 (x 1, x 2 ) for all x 1, x 2 X 1. It is well-known that Lipschitz functions are continuous.. Convexity in the space of fuzzy numbers Usually, the concept of convex set is given in relation with a vector space structure. It is known that the addition and scalar multiplication of fuzzy numbers do not form a vector space. However, since these operations are closed (see also the discussion about semilinear spaces from the Introduction) and mostly because it will be of great help later in obtaining the main results of the paper, we need the notion of convex set in the space of fuzzy numbers. Of course, the definition is exactly the same as in the case of vector spaces and therefore we have the following. Definition 1. A nonempty set Ω F(R) is called a convex subset of F(R)ifforallA, B Ω and γ [, 1] we have ((1 γ)a + γb) Ω. Remark 2. If A, B, C F(R) and γ [, 1] then taking into account the definitions of addition and scalar multiplication of fuzzy numbers, we have C = (1 γ)a + γb if and only if C L (α) = (1 γ)a L (α) + γb L (α) and C U (α) = (1 γ)a U (α) + γb U (α)forallα [, 1]. If A, B F(R) then we introduce the following subsets of F(R): [A, B] ={(1 γ)a + γb : γ [, 1]}, [A, B) ={(1 γ)a + γb : γ [, 1)}, (A, B] ={(1 γ)a + γb : γ (, 1]}, (A, B) ={(1 γ)a + γb : γ (, 1)}. We observe that a nonempty set Ω F(R) is convex if and only if [A, B] Ω for all A, B Ω. The following results are familiar or easily reachable in the case of vector topological spaces. But since the space of fuzzy numbers is not a vector space, to avoid any ambiguity and also for the uniformity of the exposure, we prefer to give the proof of each result. However, at the end of the section we will propose another approach using an embedding theorem in normed spaces. We mention that in the case of topological results we always consider the topology generated over the space of fuzzy numbers by the Euclidean distance presented in the previous section. Lemma. Let A, B F(R) and let γ 1 γ 2 γ n 1. For i {1, 2,..., n} denote C i = (1 γ i )A + γ i B. Then we have [A, B] = [A, C 1 ] [C 1, C 2 ] [C n 1, C n ] [C n, B]. Proof. We give the proof only for the case n=1 because the proof of the general case is immediate by mathematical induction. So, let γ [, 1] and denote C = (1 γ)a + γb which by Remark 2 implies that C L (α) = (1 γ)a L (α) + γb L (α), (6) C U (α) = (1 γ)a U (α) + γb U (α), α [, 1]. (7) We have to prove that [A, B] = [A, C] [C, B]. Without any loss of generality we may suppose that γ (, 1) because otherwise the proof is trivial. Firstly, we prove that [A, C] [C, B] [A, B]. For this reason let us choose arbitrarily

7 L. Coroianu / Fuzzy Sets and Systems 2 (212) D [A, C] [C, B]. If D [A, C]thenletβ [, 1] be such that D = (1 β)a+βc.thensincec = (1 γ)a+γb, and noting that the scalar multiplications involve only positive reals, we get D = (1 βγ)a + βγb. Since βγ [, 1] it easily follows that D [A, B]. The proof of the case D [C, B] is similar and therefore we omit the details. Now, we prove the converse inclusion. So, let D [A, B]. Then there exists δ [, 1] such that D = (1 δ)a + δb. This implies D L (α) = (1 δ)a L (α) + δb L (α), α [, 1]. (8) Firstly, we suppose that δ γ. Combining relations (6)and(8) weget ( D L (α) = 1 δ ) A L (α) + δ γ γ C L(α), α [, 1]. In the same manner we get that ( D U (α) = 1 δ ) A U (α) + δ γ γ C U (α), α [, 1]. Since δ/γ 1 and taking into account Remark 2, we easily get that D = (1 δ/γ)a + (δ/γ)c which implies that D [A, C]. In the remaining case, that is δ γ by similar reasonings we will obtain that D [C, B]. This last assertion completes the proof. Remark 4. We have to notice that the above proved property can be done in the case of vectorial spaces with a slightly simpler computational involvement. However, it is important that in the space of fuzzy numbers this property holds too. This will be seen later in the paper. Lemma 5. If A, B, C are fuzzy numbers such that C [A, B] then d(a, B) = d(a, C) + d(c, B). Proof. Let γ [, 1] be such that C = (1 γ)a + γb. From relations (6)to(7)weget ( ) 1/2 d(a, C) + d(c, B) = ((A L (α) C L (α)) 2 + (A U (α) C U (α)) 2 ) dα ( ) 1/2 + ((C L (α) B L (α)) 2 + (C U (α) B U (α)) 2 ) dα ( ) 1/2 = (γ 2 (A L (α) B L (α)) 2 + γ 2 (A U (α) B U (α)) 2 ) dα ( ) 1/2 + ((1 γ) 2 (A L (α) B L (α)) 2 + (1 γ) 2 (A U (α) B U (α)) 2 ) dα = γd(a, B) + (1 γ)d(a, B) = d(a, B). and the proof is complete. Lemma 6. If A, B F(R) then [A,B] is a closed set. Proof. Let (C n ) n 1 be a convergent sequence of fuzzy number with lim n C n = C and such that for all n 1we have C n [A, B]. It follows that there exists a sequence (γ n ) n 1 such that for all n 1wehaveγ n [, 1] and C n = (1 γ n )A + γ n B.WehavetoprovethatC [A, B]. Since (γ n ) n 1 is bounded it follows that there exists γ R such that γ = sup n 1 γ n. Let C = (1 γ)a + γb. Since it is easy to check that γ [, 1], it follows that

8 122 L. Coroianu / Fuzzy Sets and Systems 2 (212) C [A, B]. From the definition of γ it results the existence of a subsequence (γ kn ) n 1 of the sequence (γ n ) n 1 such that lim n γ kn = γ. Wehave d 2 (C kn, C ) = ((C kn ) L (α) C L (α))2 + ((C kn ) U (α) C U (α))2 ) dα, which by elementary calculus gives d 2 (C kn, C ) = (γ γ kn ) 2 (A L (α) B L (α)) 2 dα + (γ γ kn ) 2 (A U (α) B U (α)) 2 dα = (γ γ kn ) 2 d 2 (A, B). Since lim n γ kn = γ we immediately obtain that lim n C kn = C. On the other hand, since (C kn ) n 1 is a subsequence of the sequence (C n ) n 1 and, since lim n C n = C, it necessarily follows that lim n C kn = C. From the uniqueness of the limit we obtain C = C. Just above we have proved that C [A, B] and this implies that C [A, B]. In conclusion, we have just proved that any convergent sequence with elements in [A,B] has its limit in [A,B] and this means that the set [A,B] is a closed set. Lemma 7. Let A, B F(R), A B. Then cl((a, B)) = cl((a, B]) = cl([a, B)) = [A, B], where cl is the closure operator. Proof. First we prove that cl((a, B)) = [A, B]. Let us consider the sequences (C n ) n 1, (D n ) n 1, C n = 1 ( n A ) B, n ( D n = 1 1 ) A + 1 n n B. The definition of (A,B) implies that for all n 1wehaveC n (A, B) andd n (A, B). On the other hand it is immediate that lim n C n = B and that lim n D n = A. This implies that {A, B} cl(a, B). Now, since [A, B] = (A, B) {A, B} it necessarily follows that [A, B] cl((a, B)). But since (A, B) [A, B], it also results that cl((a, B)) cl([a, B]). The previous lemma implies that [A, B] = cl([a, B]). Consequently, we obtain cl((a, B)) [A, B] which combined with the converse inclusion gives cl((a, B)) = [A, B]. The equality cl((a, B]) = [A, B] results from the following inclusions: [A, B] = cl((a, B)) cl((a, B]) cl([a, B]) = [A, B]. By similar reasonings we will obtain that cl([a, B)) = [A, B]. In what follows, we propose another approach which leads to the same conclusions of this section, based on an embedding theorem of Rådström. It can be easily proved that the quadruple (F(R), +,, d) satisfies all the requirements of Theorem 1 in []. Applying the conclusion of Rådström s theorem if follows that (F(R), +,, d) can be embedded in a normed space ( F(R),,, D). This means that A B = A + B, λ A = λ A and D(A, B) = d(a, B) for all A, B F(R) andλ. This implies that all the convex or closed subsets of F(R) can be perceived as closed or convex subsets of the normed space F(R). More recently, Yeh proved (see [,6]) that(f(r), +,, d λ )whered λ is a weighted L 2 -type metric which generalizes the Euclidean distance d (see e.g. [7]) can be embedded in a Hilbert space. In conclusion, using either Rådström s theorem or the construction of Yeh, all the results of this section can be obtained from corresponding results from normed or even Hilbert spaces. 4. A characterization of fuzzy number-valued Lipschitz functions First we need the following.

9 L. Coroianu / Fuzzy Sets and Systems 2 (212) Lemma 8. Let A, B F(R), A B. Furthermore, we consider the family of closed convex subsets of F(R), F ={Ω i : i {1, 2,..., n}} such that [A, B] n Ω i. i=1 Then, there exist k {1, 2,..., n}, {C j : j {, 1,..., k}} [A, B], with C = A and C k = B respectively, and {Ω l j : j {1, 2,..., k}} F, such that: (i) ) [A, B] = k j=1 [C j 1, C j ]; (ii) d(a, B) = k j=1 d(c j 1, C j ); (iii) [C j 1, C j ] Ω l j for all j {1, 2,..., k}. Proof. We prove the lemma by mathematical induction on the number of elements of the family F denoted in what follows with card(f). If card(f) = 1 then for k=1, C = A, C 1 = B, it is immediate that the conclusions of the lemma hold. Now, suppose that card(f) = n, n 2 and suppose that the conclusions of the lemma hold in the case of families with n 1 elements. Since [A, B] n i=1 Ω i, it follows that there exists i 1 {1, 2,..., n} such that A Ω i1. If B Ω i1 then owing to the convexity of Ω i1, it follows that [A, B] Ω i1 which means that the conclusions of the lemma hold by choosing k=1, C = A, C 1 = B, andω l1 = Ω i1. For this reason, in all what follows we will suppose that B / Ω i1. Let and let γ = inf{γ [, 1] : (γa + (1 γ)b) [A, B] Ω i1 } (1) (9) C = γ A + (1 γ )B. (11) Clearly, we have C [A, B]. Moreover, we will prove that C Ω i1 too. From the definition of γ results the existence of a sequence (γ n ) n 1 such γ n 1, lim n γ n = γ and (γ n A + (1 γ n )B) [A, B] Ω i1 for all n 1. For each n 1, denoting D n = γ n A + (1 γ n )B we obtain d 2 (D n, C) = (γ γ n ) 2 (A L (α) B L (α)) 2 dα + (γ γ n ) 2 (A U (α) B U (α)) 2 dα = (γ γ n ) 2 d 2 (A, B), which immediately implies that lim n D n = C. Since for all n 1wehaveD n [A, B] Ω i1 and since [A, B] Ω i1 is a closed subset of F(R) as an intersection of closed subsets from F(R), it follows that C [A, B] Ω i1.now,since {A, C} [A, B] Ω i1 and since [A, B] Ω i1 is a convex subset of F(R) as an intersection of convex subsets of F(R), we conclude that [A, C] [A, B] Ω i1. Consequently, by Lemma (case n=1) we get that [A, B] = [A, C] [C, B] and in addition by Lemma 5 we have d(a, B) = d(a, C) + d(c, B). (12) If C=B then the conclusions of the lemma hold by choosing k=1, C = A, C 1 = B, andω l1 = Ω i1. Therefore, in all that follows we may suppose that C B. This fact combined with relation (11) implies that γ >, information that will be very useful later. Next we prove that (C, B] ([A, B] Ω i1 ) =. By way of contradiction suppose that there exists D (C, B] ([A, B] Ω i1 ). Then let β [, 1] be such that D = βc + (1 β)b. (1) If β = thend = B which implies B [A, B] Ω i1,thatisb Ω i1, a contradiction since we have assumed that B / Ω i1.ifβ = 1 then we easily get that D = C and this is a contradiction since D (C, B]. Summarizing, we necessarily have that β (, 1). Combining relations (11)and(1) we obtain that (again, it is important that the scalar multiplication operations involve only positive reals) D = βγ A + (1 βγ )B.

10 124 L. Coroianu / Fuzzy Sets and Systems 2 (212) On the other hand, since γ > andβ (, 1), it follows that βγ < γ and this contradicts with the definition of γ (see relation (1)) since we have assumed that D [A, B] Ω i1. Thus, we obtain that (C, B] ([A, B] Ω i1 ) =. Taking into account relation (9), it follows that (C, B] i I Ω i where I ={1, 2,..., n} \{i 1 }. This implies that cl((c, B]) cl( i I Ω i) which by Lemma 7 and by the fact that i I Ω i is a closed set as finite union of some closed sets, implies that [C, B]) i I Ω i. Considering the family of n 1elementsF 1 ={Ω i : i {1, 2,..., n}\{i 1 }}, according to our assumption it follows that there exist k {1, 2,..., n 1}, {C j : j {, 1,..., k} [C, B], with C = C and C k = B respectively, and {Ω p j ; j {1, 2,..., k}} F 1, such that: (i) [C, B] = k j=1 [C j 1, C j ]; (ii) d(c, B) = k j=1 d(c j 1, C j ); (iii) [C j 1, C j ] Ω p j for all j {1, 2,..., k}. Take C = A, C j+1 = C j for j {, 1,..., k}, Ω l 1 = Ω i1, Ω l j+1 = Ω p j for j {1, 2,..., k}. From the properties of C j, j {, 1,..., k + 1} and Ω l j, j {1, 2,..., k + 1}, it is immediate that assertions (i) and (iii) of the present lemma hold. Then, since d(a, B) = d(a, C) + d(c, B) and since d(c, B) = k j=1 d(c j 1, C j ), it is immediate that assertion (ii) of the lemma holds too. This completes the proof. From the previous result we obtain the following main result of the paper. Theorem 9. Let F ={Ω i : i {1, 2,..., n}} be a family of closed convex subsets of F(R) such that F(R) = n Ω i. i=1 Furthermore, let f : F(R) F(R) denote a function with the property that there exist the positive real constants, c i, i {1, 2,..., n} such that for all i {1, 2,..., n} and A, B Ω i we have (14) d( f (A), f (B)) c i d(a, B). (15) Then d( f (A), f (B)) cd(a, B), A, B F(R), where c = max{c i : i {1, 2,..., n}}. Proof. Let us choose arbitrarily A, B F(R), A B (the case A=B is trivial). Relation (14) implies [A, B] n i=1 Ω i. Applying Lemma 8, it follows that there exist k {1, 2,..., n}, {C j : j {, 1,..., k}} [A, B], with C = A and C k = B respectively, and {Ω l j : j {1, 2,..., k}} F, such that assertions (i) (iii) of Lemma 8 hold. First of all we have d( f (A), f (B)) k d( f (C j 1 ), f (C j )). (16) j=1 Then, from assertion (iii) of Lemma 8, we obtain {C j 1, C j } Ω l j, j {1, 2,..., k} which by relation (15) implies d( f (C j 1 ), f (C j )) c l j d(c j 1, C j ). Since for each i {1, 2,..., n} we have c i c, it follows that for each j {1, 2,..., k} we have d( f (C j 1 ), f (C j )) cd(c j 1, C j ). Replacing in inequality (16) we obtain d( f (A), f (B)) c k d(c j 1, C j ) j=1 and since assertion (ii) of Lemma 8 implies d(a, B) = k j=1 d(c j 1, C j ), we get that d( f (A), f (B)) cd(a, B)and we obtain the desired conclusion.

11 L. Coroianu / Fuzzy Sets and Systems 2 (212) Remark 1. The above proved theorem says that if the space of fuzzy numbers is covered by a finite family of closed convex subsets and a function f : F(R) F(R) is Lipschitz relatively to each subset of the family then it is Lipschitz on the whole space of fuzzy numbers. This result will be very useful in finding the best Lipschitz constant of the trapezoidal approximation operator preserving the value and the ambiguity as it will be seen in the next section. We end this section by mentioning that we do not exclude the possibility of obtaining the proofs of Corollary 8 and Theorem 9 respectively, from corresponding results in normed spaces, using again the embedding theorem of Rådström. 5. Best Lipschitz constant of the trapezoidal approximation operator preserving the value and ambiguity The algorithms to compute the trapezoidal approximation (with respect to the Euclidean distance) of a fuzzy number which preserves the value and the ambiguity are given in Theorem 7 and Corollary 8 in the paper [1]. These results are obtained as solutions of minimization problems in the Euclidean vector space R 2. The so-called extended trapezoidal fuzzy numbers introduced by Yeh in the paper [4] are used. The trapezoidal approximation is given on cases and it is obtained by solving the minimization problem given in relations (25) () of the paper [1]. There are four possible algorithms to compute the proper trapezoidal approximation of a fuzzy number which preserves the value and ambiguity, each one corresponding to a nonempty subset of F(R) and these subsets form disjoint pairs. But a careful inside in the proof of Theorem 5 from page 184 (see also the geometrical interpretation of the minimization problem illustrated in Fig. 1 of the discussed paper) proves that the formulas for computing the trapezoidal approximation described in cases (i) (iv) just below the end of the proof of Theorem 5 remain valid if in all the inequalities from cases (i) (iv) we consider non-strict inequalities. Therefore, in Theorem 7 and Corollary 8 we can replace all the strict inequalities with non-strict inequalities. Taking into account this information we give a slightly modified version of Corollary 8 as follows. Theorem 11. Let A F(R), A α = [A L (α), A U (α)], α [, 1], and let T (A) = (t 1, t 2, t, t 4 ) denote the nearest trapezoidal fuzzy number to A which preserves the value and the ambiguity. (i) If then (α 1)A L (α) dα (α 1)A U (α) dα (17) t 1 = t 2 = t = t 4 = (4 6α)A L (α) dα, (6α 2)A L (α) dα, (6α 2)A U (α) dα, (4 6α)A U (α) dα. (18) (19) (2) (21) (ii) If (α 1)A L (α) dα + (α 1)A U (α) dα (22)

12 126 L. Coroianu / Fuzzy Sets and Systems 2 (212) then then and then t 1 = 6 t 2 = t = t 4 = 2 (iii) If αa L (α) dα 4 (α 1)A L (α) dα + t 1 = t 2 = t = 2 t 4 = 6 (iv) If αa U (α) dα. αa U (α) dα 4 (α 1)A L (α) dα (α 1)A L (α) dα + (α 1)A L (α) dα + t 1 = 2 t 2 = t = t 4 = A L (α) dα αa L (α) dα, αa U (α) dα, (2) (α 1)A U (α) dα (25) (α 1)A L (α) dα + (2 6α)A L (α) dα + 2 (24) (26) αa L (α) dα. (27) (α 1)A U (α) dα, (28) (α 1)A U (α) dα (29) (α 1)A U (α) dα () (6α 2)A U (α) dα, (1) (α 1)A U (α) dα, (2) A U (α) dα. () The following auxiliary result will help us in the main application of this section. Lemma 12. Let (A n ) n N,(A n ) α = [(A n ) L (α), (A n ) U (α)], α [, 1] be a sequence in the space F(R) such that lim n A n = A, where Then A α = [A L (α), A U (α)], α [, 1]. lim n α k (A n ) L (α) dα = α k A L (α) dα

13 and lim n for all k N. α k (A n ) U (α) dα = L. Coroianu / Fuzzy Sets and Systems 2 (212) α k A U (α) dα Proof. Let ε >. Because lim n A n = A, there exists n ε N such that d 2 (A n, A) = for every n n ε.weget (((A n ) L (α) A L (α)) 2 + ((A n ) U (α) A U (α)) 2 ) dα < ε 2 ((A n ) L (α) A L (α)) 2 dα < ε 2 for every n n ε and ((A n ) U (α) A U (α)) 2 dα < ε 2 for every n n ε. Let k N.Then α k (A n ) L (α) dα α k 1 A L (α) dα α k ((A n ) L (α) A L (α)) dα α 2k ((A n ) L (α) A L (α)) 2 dα for every n n ε,thatis ((A n ) L (α) A L (α)) 2 dα < ε lim n α k (A n ) L (α) dα = α k A L (α) dα. The proof is similar in the case of the second limit and therefore we omit the details. We present now the main result of this section. Theorem 1. If T : F(R) F T (R) is the trapezoidal approximation operator which preserves the value and the ambiguity given in Theorem 11, then d(t (A), T (B)) d(a, B) for all A, B F(R) and the value ( )/ is the best possible Lipschitz constant of the operator T. Proof. Let us introduce the sets Ω 1 ={A F(R) :A satisfies condition (17)}, Ω 2 ={A F(R) :A satisfies condition (22)}, Ω ={A F(R) :A satisfies condition (25)}, Ω 4 ={A F(R) :A satisfies conditions (28).()}. It is elementary to verify the convexity of the sets Ω i, i {1, 2,, 4}. Then from Lemma 12 it follows that each one of the sets Ω i, i {1, 2,, 4}, is a closed set. We exemplify with the set Ω 1 because the proof in the remaining

14 128 L. Coroianu / Fuzzy Sets and Systems 2 (212) cases is similar. Let (A n ) n 1,(A n ) α = [(A n ) L (α), (A n ) U (α)], α [, 1] be a sequence in the space F(R) such that lim n A n = A and such that for all n 1wehaveA n Ω 1. As a first implication we have (α 1)(A n ) L (α) dα (α 1)(A n ) U (α) dα. Passing to limit with n and taking into account Lemma 12, it follows that (α 1)A L (α) dα (α 1)A U (α) dα, that is A Ω 1 and clearly this implies that Ω 1 is a closed subset of F(R). As a first conclusion it follows that the space F(R) is covered by the family of closed convex subsets of F(R), F ={Ω i : i {1, 2,, 4}}. Weintendto apply Theorem 9 with respect to the family F and therefore, in what follows we will determine the Lipschitz constants of the operator T relatively to the sets of the family F. Thus, for A, B F(R) with T (A) = (t 1, t 2, t, t 4 )and T (B) = (t 1, t 2, t, t 4 ), we distinguish the following four cases: (i) A, B Ω 1. It is immediate that the case (i) in Theorem 11 is applicable to compute T(A) andt(b). We observe that formulas (18) (21) coincide with the formulas for computing the nearest extended trapezoidal approximation of Grzegorzewski and Mrówka (see e.g. [4], the formulas from the first four lines from the top on page 1 or Theorem in [7]). Therefore, in this case we have T (A) = T e (A)andT(B) = T e (B)whereT e (A)andT e (B) denote the extended trapezoidal approximations of A and B respectively. Since by Proposition 4.4 in [2] we have d(t e (A), T e (B)) d(a, B), we obtain in this case the relation d(t (A), T (B)) d(a, B). (4) (ii) A, B Ω 2. It is immediate that the case (ii) in Theorem 11 is applicable to compute T(A) andt(b). Noting formula (5) and since by relations (2) (24)wehavet 2 = t = t 4 and t 2 = t = t 4, we obtain d 2 (T (A), T (B)) = 1 (t 1 t 1 )2 + 4 (t 2 t 2 )2 + 1 (t 1 t 1 )(t 2 t 2 ), which again by formulas (2) (24), after some simple calculations give ( 2 ( d 2 (T (A), T (B)) = 12 α(a L (α) B L (α)) dα) + 8 α(a U (α) B U (α)) dα ( )( ) 12 α(a L (α) B L (α)) dα α(a U (α) B U (α)) dα. Now, applying the Schwarz inequality we get d 2 (T (A), T (B)) 12 (A L (α) B L (α)) 2 dα α 2 dα + 8 (A U (α) B U (α)) 2 dα +12 (A L (α) B L (α)) 2 dα α 2 dα (A U (α) B U (α)) 2 dα = 4 (A L (α) B L (α)) 2 dα + 8 (A U (α) B U (α)) 2 dα ) 2 α 2 dα α 2 dα +4 (A L (α) B L (α)) 2 dα (A U (α) B U (α)) 2 dα. (5) Let m be such that (A U (α) B U (α)) 2 dα = m (A L (α) B L (α)) 2 dα. (6)

15 L. Coroianu / Fuzzy Sets and Systems 2 (212) Using the above equality in relation (5)weget ( d 2 (T (A), T (B)) 4 + 8m ) + 4 m (A L (α) B L (α)) 2 dα Noting that (m + 1) it follows that = 4 + 8m + 4 m m + 1 (A L (α) B L (α)) 2 dα = m = (m + 1) (A L (α) B L (α)) 2 dα. (A L (α) B L (α)) 2 dα + (A U (α) B U (α)) 2 dα + (A L (α) B L (α)) 2 dα (A L (α) B L (α)) 2 dα = d 2 (A, B), d 2 (T (A), T (B)) 4 + 8m + 4 m d 2 (A, B). (7) m + 1 Let us introduce the function f :[, ) R, f (m) = 4 + 8m + 4 m. m + 1 We have f (m) = ( 6m 4 m + 6)/( m(m + 1) 2 ). It is immediate that f ((( 1 + 1)/) 2 ) =. Then it is easy to check that f is strictly increasing on [, (( 1 + 1)/) 2 ] and strictly decreasing on ((( 1 + 1)/) 2, ). This implies that m = (( 1 + 1)/) 2 is a global maximum point of the function f. Since by simple calculations we get f ((( 1 + 1)/) 2 ) = ( )/, introducing the maximum value in relation (7)weget d(t (A), T (B)) d(a, B). (8) (iii) A, B Ω. To avoid laborious calculus we propose a different approach. We easily observe that A and B both belong to the set Ω 2. Then, having in mind the scalar multiplication of fuzzy numbers presented in the first section, one can easily prove that T ( A) = T (A)andT ( B) = T (B). This is not surprising since it can easily be proved that the operator T is scale invariant. On the other hand, noting that in general for two arbitrarily chosen fuzzy numbers C and D we have d(c, D) = d( C, D), it follows that d(t (A), T (B)) = d( T (A), T (B)) = d(t ( A), T ( B)) and since A and B both belong to the set Ω 2, by relation (8) we obtain d(t ( A), T ( B)) d( A, B) = d(a, B) and finally, we have d(t (A), T (B)) d(a, B). (9) (iv) A, B Ω 4. It is immediate that case (iv) in Theorem 11 is applicable to compute T(A)andT(B). Noting formula (5) and since by relations (1) ()wehavet 2 = t and t 2 = t, we obtain d 2 (T (A), T (B)) = 1 (t 1 t 1 )2 + 2 (t 2 t 2 )2 + 1 (t 4 t 4 )2 + 1 (t 2 t 2 )[(t 1 t 1 ) + (t 4 t 4 )]. (4)

16 1 L. Coroianu / Fuzzy Sets and Systems 2 (212) We introduce the following notations: a 1 = a 2 = a = a 4 = (A L (α) B L (α)) dα, (A U (α) B U (α)) dα, Relations (1) ()imply (α 1)(A L (α) B L (α)) dα, (α 1)(A U (α) B U (α)) dα. t 1 t 1 = 2a 1 2a 4, t 2 t 2 = a + a 4, t 4 t 4 = 2a 2 2a. Using these equalities in relation (4) we obtain d 2 (T (A), T (B)) = 4 (a 1 a 4 ) (a + a 4 ) (a 2 a ) (a + a 4 )(a 1 a 4 + a 2 a ), which by simple calculations give d 2 (T (A), T (B)) = 4 (a2 1 + a2 2 + a2 + a2 4 ) + 2 (a 1a + a 2 a 4 ) 2(a 1 a 4 + a 2 a ). Now, using the obvious inequalities a 1 a + a 2 a (a2 1 + a2 2 + a2 + a2 4 )and a 1a 4 + a 2 a 1 2 (a2 1 + a2 2 + a2 + a2 4 ), we easily obtain d 2 (T (A), T (B)) 8 (a2 1 + a2 2 + a2 + a2 4 ). Then, replacing a i, i {1, 2,, 4}, with their initial values and then using the Schwarz inequality, we get [ ( d 2 (T (A), T (B)) ( ) 2 ] (A L (α) B L (α)) dα) + (A U (α) B U (α)) dα + 8 ( ) 2 (α 1)(A L (α) B L (α)) dα + 8 ( ) 2 (α 1)(A U (α) B U (α)) dα 8 [ ] (A L (α) B L (α)) 2 dα + (A U (α) B U (α)) 2 dα + 8 (α 1) 2 dα (A L (α) B L (α)) 2 dα + 8 (α 1) 2 dα (A U (α) B U (α)) 2 dα = 16 [ (A L (α) B L (α)) 2 dα + Summarizing, in this last case we obtain ] (A U (α) B U (α)) 2 dα = 16 d2 (A, B). (41) d(t (A), T (B)) 4 d(a, B). (42)

17 L. Coroianu / Fuzzy Sets and Systems 2 (212) Now we return to the purpose of this theorem, that is to find the best Lipschitz constant of the operator T. Analyzing cases (i) (iv) we conclude that thelipschitz constants corresponding to the elements of the family F ={Ω i : i {1, 2,, 4}} are c 1 = 1, c 2 = c = ( )/ andc 4 = 4 (see relations (4), (8), (9) and(42)). Applying Theorem 9 and noting that max{c i : i {1, 2,, 4}} = ( )/, it follows that for all A, B F(R)wehave d(t (A), T (B)) d(a, B). To prove that the value ( )/ is the best Lipschitz constant of the operator T we need to find A, B F(R), A B, such that d(t (A ), T (B )) = d(a, B ). (4) Looking over relation (5), in order to obtain equality instead of inequality, we need to find A, B Ω 2 with B of the form (B ) L (α) = (A ) L (α) + k 1 α, (B ) U (α) = (A ) U (α) k 2 α, α [, 1], where k 1 >, k 2 >. Actually, it is not necessary to impose the conditions of strict positivity for k 1 and k 2 but in this way, provided that (A ) L (1) + k 1 (A ) U (1) k 2, we get that B is a proper fuzzy number. Moreover, it is elementary to check that if A Ω 2 then B Ω 2 too when k 1 > andk 2 >. Then, A and B have to satisfy relation (6)for m = (( 1 1)/) 2, which immediately implies that ( 1 ) 1 k 1 k 2 =. Inspired by the above considerations we introduce the fuzzy numbers A, B, (A ) L (α) = 9 α + 1, (A ) U (α) = 96, 1 1 (B ) L (α) = (A ) L (α) + α, (B ) U (α) = (A ) U (α) α, α [, 1]. It is immediate that A, B Ω 2 and therefore applying formulas (2) (24) we obtain T (A ) = (27, 96, 96, 96) and ( T (B ) = ( 1 1) 9, 96 2( 1 1) 9 ), 96 2( 1 1), 96 2( 1 1). 9 9 Now, it is easy to verify that relation (4) holds. This last assertion ends the proof. From the previous theorem it results that the value ( )/ is the best Lipschitz constant of the operator T applied for the entire set of fuzzy numbers. Still, since the operator T is given on four cases, it would be of some interest to determine the best Lipschitz constant for each case in part. That is to find the best Lipschitz constant for each one of the sets Ω i, i {1, 2,, 4}. Clearly, from the example given at the end of the proof of the previous theorem it follows that the value ( )/ is the best Lipschitz constant applied for the set Ω 2. Then, for A and B from the discussed example we observe that A and B both belong to Ω. Noting again that the operator T is scale invariant we easily obtain that (4) holds if we replace A and B with A and B respectively. This means

18 12 L. Coroianu / Fuzzy Sets and Systems 2 (212) that ( )/ is the best Lipschitz constant applied for the set Ω. Moreover, if A and B are trapezoidal fuzzy numbers then it is immediate that (17) holds and therefore we have A, B Ω 1. In addition, since the operator T fulfils the identity criteria it follows that T (A) = A and T (B) = B. This fact combined with inequality (4) proves that the value 1 is the best Lipschitz constant applied for the set Ω 1. Unfortunately, in the case of the set Ω 4 we cannot find A, B Ω 4, A B such that relation (42) would become equality. By way of contradiction let us suppose that this possibility exits. Because in relation (41) we used the inequalities ( 2 (A L (α) B L (α)) dα) (A L (α) B L (α)) 2 dα, ( 2 (A U (α) B U (α)) dα) ( 2 (α 1)(A L (α) B L (α)) dα) ( 2 (α 1)(A U (α) B U (α)) dα) (A U (α) B U (α)) 2 dα, (α 1) 2 dα (α 1) 2 dα (A L (α) B L (α)) 2 dα, (A U (α) B U (α)) 2 dα, it follows that all these inequalities become equalities and therefore, in what follows we consider them as equalities. From the first and the third equality it results the existence of the real constants k 1 and k 2 such that and A L (α) B L (α) = k 1 A L (α) B L (α) = k 2 (α 1) almost everywhere α [, 1]. Clearly, the above two equalities are possible only when k 1 = k 2 = and so we get that A L (α) = B L (α) almost everywhere α [, 1]. Analogously, from the second and the fourth equality we get that A U (α) = B U (α) almost everywhere α [, 1] which combined with the previous relation implies that A=B (see the discussion about the equality of two fuzzy numbers given in Section 2) and this clearly contradicts our assumption. In conclusion, the best Lipschitz constant applied for the set Ω 4 remains an open question. 6. Some applications In this last section we will apply the estimation obtained in Theorem 1 to calculate the trapezoidal approximation of a fuzzy number preserving the value and ambiguity within a reasonable error in the case when the direct algorithm is not applicable. We will consider the same example as in the case of the trapezoidal approximation operator preserving the expected interval (see [16, Example ]). Then we will improve the estimation on the defect of additivity of the trapezoidal approximation operator preserving the value and the ambiguity, estimation which was obtained in the paper [1]. Example 14. Let us consider the fuzzy number A, A L (α) = e α2, A U (α) = 4 α, α [, 1]. We will determine T(A) with an error less than 1 2 with respect to the Euclidean metric d. For this purpose let us consider the sequence of fuzzy numbers (A n ) n 1,(A n ) L (α) = 1 + α 2 + α 4 /2! + α 6 /! + +α 2n /n!, A U (α) = 4 α, α [, 1]. Following the same root as in the case of the trapezoidal approximation operator preserving the expected interval we get that d 2 e 2 (A n, A) ((n + 1)!) 2, n 1. (4n + 5) Applying Theorem 1 we obtain d 2 (T (A n ), T (A)) e 2 ((n + 1)!) 2 (4n + 5).

19 L. Coroianu / Fuzzy Sets and Systems 2 (212) Obviously, for n 5wehave d(t (A n ), T (A)) 1 2. For n=5 case (i) in Theorem 11 is applicable to compute the trapezoidal approximation of A 5 preserving the value and the ambiguity, therefore T (A 5 ) = (.69595, ,, 4) and we thus obtained an approximation of T(A) with an error less than 1 2. Note that in the case of the trapezoidal approximation preserving the expected interval, since the best Lipschitz 5 constant is the value, we obtained the desired approximation for n=4. It is known by Example 24 in the paper [1] that the trapezoidal approximation operator preserving the value and the ambiguity is not additive. Following the ideas in [6] the notion of defect of additivity of a trapezoidal approximation operator was introduced in the paper [1]. Definition 15 (Ban et al. [1, Definition 25]). Let A F(R) andt : F(R) F T (R) be a trapezoidal approximation operator with respect to a distance denoted with D. The defect of additivity of the operator t with respect to fuzzy number A is given by δ t,d (A) = sup D(t(A) + t(b), t(a + B)). B F(R) To obtain an estimation for δ T,d (A) in the case of the trapezoidal approximation preserving the value and ambiguity we need the following. Lemma 16. Let t : F(R) F T (R) be a trapezoidal approximation operator with respect to a distance denoted with D. Suppose that the following requirements hold: (i) t(o) = O where O is the trapezoidal fuzzy number (,,,); (ii) there exists a positive real constant c such that D(t(A), t(b)) cd(a, B), A, B F(R); (iii) we have D(A + C, B + C) = D(A, B) for all A, B, C F(R). Then δ t,d (A) 2cD(O, A) for all A F(R). Proof. Let us choose arbitrarily a fuzzy number A. Then, for B F(R) wehave D(t(A) + t(b), t(a + B)) D(t(A) + t(b), t(b)) + D(t(B), t(a + B)) D(t(A), O) + cd(b, A + B) = D(t(A), t(o)) + cd(o, A) cd(o, A) + cd(o, A) = 2cD(O, A) and the proof is complete.

20 14 L. Coroianu / Fuzzy Sets and Systems 2 (212) Corollary 17. Let A be a fuzzy number. If T denotes the trapezoidal approximation operator preserving the value and the ambiguity, then ( 1 1 ) 1/2 δ T,d (A) 2 (A 2 L (α) + A2 U (α)) dα. Proof. Firstly, we notice that the operator T and the Euclidean distance d fulfil the requirements of Lemma 16.Nowthe proof is immediatefromthe above lemma, replacingc with the best Lipschitz constant of the trapezoidal approximation operator preserving the value and ambiguity, that is c = ( )/, and noting that ( ) 1/2 d(o, A) = (A 2 L (α) + A2 U (α)) dα. 7. Conclusions In this paper we have proved that many important properties concerning convex subsets in normed spaces remain valid in the space of fuzzy numbers. These properties were used to obtain a characterization of Lipschitz fuzzy valued functions (Theorem 9), which is useful in proving that a trapezoidal approximation operator satisfies the Lipschitz condition and in the case of the trapezoidal approximation preserving the value and ambiguity even the best Lipschitz constant was found. The method is suitable to be applied for most of the triangular trapezoidal or parametric approximation operators that can be found in the literature since these operators fulfil the requirements of Theorem 9. Future work will be dedicated to find better estimates for the defect of additivity of fuzzy approximation operators. We have already started a research in the case of the trapezoidal approximation operator preserving the expected interval and the results are encouraging. Acknowledgments This work was possible with the financial support of the Sectoral Operational Programme for Human Resources Development 27 21, co-financed by the European Social Fund, under the project number POSDRU/17/1.5/S/ with the title Modern Doctoral Studies: Internationalization and Interdisciplinarity and, with the support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE References [1] S. Abbasbandy, M. Amirfakhrian, The nearest trapezoidal form of a generalized left right fuzzy number, Int. J. Approx. Reason. 4 (26) [2] S. Abbasbandy, M. Amirfakhrian, The nearest approximation of a fuzzy quantity in parametric form, Appl. Math. Comput. (New York) 172 (26) [] S. Abbasbandy, B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Appl. Math. Comput. (New York) 156 (24) [4] S. Abbasbandy, T. Hajjari, Weighted trapezoidal approximation-preserving cores of a fuzzy number, Comput. Math. Appl. 59 (21) [5] T. Allahviranloo, M. Adabitabar Firozja, Note on Trapezoidal approximation of fuzzy numbers, Fuzzy Sets Syst. 158 (27) [6] A.I. Ban, S.G. Gal, Defects of Properties in Mathematics. Quantitative Characterizations, World Scientific, New Jersey, 22. [7] A.I. Ban, Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval, Fuzzy Sets Syst. 159 (28) [8] A.I. Ban, On the nearest parametric approximation of a fuzzy number-revisited, Fuzzy Sets Syst. 16 (29) [9] A.I. Ban, Trapezoidal and triangular approximations of fuzzy numbers inadvertences and corrections, Fuzzy Sets Syst. 16 (29) [1] A.I. Ban, A. Brândaş, L. Coroianu, C. Negruţiu, O. Nica, Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the value and ambiguity, Comput. Math. Appl. 61 (211) [11] A.I. Ban, L. Coroianu, Continuity and linearity of the trapezoidal approximation preserving the expected interval operator, in: IFSA-EUSFLAT World Congress, 2 24 July 29, pp