Fuzzy vectors via convex bodies

Cheng-Yong Dua, Lili Shenb,∗

aSchool of Mathematical Sciences and V. C. & V. R. Key Lab, Sichuan Normal University, Chengdu 610068, ChinabSchool of Mathematics, Sichuan University, Chengdu 610064, China

Abstract

In the most accessible terms this paper presents a convex-geometric approach to the study of fuzzy vectors. Motivatedby several key results from the theory of convex bodies, we establish a representation theorem of fuzzy vectors throughsupport functions, in which a necessary and sufficient condition for a function to be the support function of a fuzzyvector is provided. As applications, symmetric and skew fuzzy vectors are postulated, based on which a Mares core ofeach fuzzy vector is constructed through convex bodies and support functions, and it is shown that every fuzzy vectorover the n-dimensional Euclidean space has a unique Mares core if, and only if, the dimension n = 1.

Keywords: fuzzy vector, convex body, support function, symmetric fuzzy vector, skew fuzzy vector, Mares core,Mares equivalence2020 MSC: 26E50, 03E72, 52A20

1. Introduction

Since Zadeh introduced the concept of fuzzy sets [29] in the 1960s, fuzzy numbers, as a special kind of fuzzysubsets of the set R of real numbers, have received considerable attention both in the theory and the applications offuzzy sets [6, 7, 8, 9, 10, 11, 5, 4]. The notion of fuzzy number may be generalized to fuzzy vector (also n-dimensionalfuzzy number) without obstruction, simply by replacing R with the n-dimensional Euclidean space Rn in its definition,which has been widely studied as well [16, 24, 3, 31, 32, 28, 27, 19].

Convex geometry, as an independent branch of mathematics, has a much longer history that dates back to the turnof the 20th century [1], in which several contributions can be even traced back to the ancient works of Euclid andArchimedes. As a well-developed theory in the past decades, convex geometry has been applied to different areas ofgeometry, analysis and computer science [14, 18].

It is well known that fuzzy vectors can be characterized through their level sets (see Theorem 2.2.3, originatedfrom [22, 17]). In particular, each level set of a fuzzy vector is a nonempty, compact and convex subset of Rn, whichis precisely a convex body [13, 26] in the sense of convex geometry. It is then natural to consider the possibility ofexploiting the powerful arsenal of convex geometers in the realm of fuzzy vectors, and it is the motivation of this paper.Being tailored to the readership of fuzzy set theorists, the geometric machinery involved in this paper are presented inthe most accessible terms, so that hopefully, even a reader who is not familiar with the extensive apparatus of convexgeometry could easily follow up.

Specifically, inspired by several key results from the theory of convex bodies, this paper is intended to representfuzzy vectors through support functions (Section 2) and, as applications, investigate Mares cores of fuzzy vectors(Section 3). The backgrounds and our main results are illustrated as follows.

∗Corresponding author.Email addresses: cyd9966@hotmail.com (Cheng-Yong Du), shenlili@scu.edu.cn (Lili Shen)

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1.1. Representation of fuzzy vectors via support functions

Support functions play an essential role in the study of fuzzy vectors. Explicitly, the support function [24, 3] of afuzzy vector u is given by

hu : [0, 1] × S n−1 → R, hu(α, x) :=∨t∈uα

〈t, x〉,

where S n−1 is the unit sphere in Rn, 〈−,−〉 refers to the standard Euclidean inner product of Rn, and uα is the α-levelset of u. The following question arises naturally:

Question 1.1.1. Can we find a necessary and sufficient condition for a function

h : [0, 1] × S n−1 → R

to be the support function of a (unique) fuzzy vector?

This question is partially answered by Zhang–Wu in [32], where several sufficient conditions are provided, thoughneither of them is necessary. In order to fully solve this question, several key results from the theory of convex bodiesare exhibited in Subsection 2.1:

• There exists a hyperplane that supports a convex body at any of its boundary point (Theorem 2.1.3).

• A function h : S n−1 → R is the support function of a (unique) convex body if, and only if, it is sublinear(Theorem 2.1.5).

Without assuming any a-priori background by the reader on convex geometry, in Subsection 2.1 we develop allneeded ingredients from scratch for the self-containment of this paper. Then, based on the well-known characteriza-tion of fuzzy vectors through convex bodies (Theorem 2.2.3), a representation theorem of fuzzy vectors via supportfunctions is established (Theorem 2.2.4). Explicitly, it is shown in Theorem 2.2.4 that a function

h : [0, 1] × S n−1 → R

is the support function of a (unique) fuzzy vector if, and only if,

(VS1) h(α,−) : S n−1 → R is sublinear for each α ∈ [0, 1],

(VS2) h(−, x) : [0, 1]→ R is non-increasing, left-continuous on (0, 1] and right-continuous at 0 for each x ∈ S n−1.

Therefore, a perfect answer is provided for Question 1.1.1.

1.2. Mares cores of fuzzy vectors

In the recent works of Qiu–Lu–Zhang–Lan [25] and Chai–Zhang [2], a crucial property of fuzzy numbers regard-ing their Mares cores is revealed. Explicitly, a fuzzy number u is skew [2] if it cannot be written as the sum of a fuzzynumber and a non-trivial symmetric fuzzy number in the sense of Mares [20]; that is, if

u = v ⊕ w

and w is symmetric, then w is constant at 0. A fuzzy number v is the Mares core [21, 15] of a fuzzy number u if v isskew and u = v ⊕ w for some symmetric fuzzy number w. The following theorem combines the main results of [25]and [2]:

Theorem 1.2.1. Every fuzzy number has a unique Mares core, so that every fuzzy number can be decomposed in aunique way as the sum of a skew fuzzy number, given by its Mares core, and a symmetric fuzzy number.

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It is then natural to ask whether it is possible to establish the n-dimensional version of Theorem 1.2.1 for generalfuzzy vectors. Unfortunately, a negative answer will be given in Section 3 (Theorem 3.4.10).

Based on the representation of fuzzy vectors through convex bodies and support functions in Section 2, Theorem3.1.1 describes the sum of fuzzy vectors defined by Zadeh’s extension principle through the Minkowski sum of convexbodies and the sum of their support functions, which is the cornerstone of the results of Section 3. Then, in Subsection3.2, the notion of symmetric fuzzy vector is defined as symmetric around the origin in accordance with the case ofn = 1 (cf. [25, Remark 2.1]); using the language of convex bodies and support functions, a fuzzy vector is symmetricwhenever its level sets are closed balls centered at the origin, or whenever the support functions of its level sets areconstant (Theorem 3.2.3).

The notion of symmetric fuzzy vector allows us to postulate skew fuzzy vectors and Mares cores of fuzzy vectorsin Subsection 3.4. However, for the purpose of studying their properties we have to be familiar with the inner parallelbodies of convex bodies, and this is the subject of Subsection 3.3, in which we characterize inner parallel bodiesthrough support functions, and prove that every convex body can be uniquely decomposed as the Minkowski sum ofan irreducible convex body and a closed ball centered at the origin (Theorem 3.3.4).

The first part of Subsection 3.4 is devoted to the decomposition

u = c(u) ⊕ s(u) (1.i)

of each fuzzy vector u ∈ Fn, where c(u) is a Mares core of u, and s(u) is a symmetric fuzzy vector (Theorem 3.4.5).However, unlike Theorem 1.2.1 for the case of n = 1, Example 3.4.9 reveals that Equation (1.i) may not be the uniqueway of decomposing a fuzzy vector. Hence, we indeed obtain a negative answer to the possibility of establishing then-dimensional version of Theorem 1.2.1, which is stated as Theorem 3.4.10.

Finally, we investigate Mares equivalent fuzzy vectors in Subsection 3.5. As we shall see, comparing with Maresequivalent fuzzy numbers (see Corollary 3.5.6), the Mares equivalence relation of fuzzy vectors may behave in quitedifferent ways. As Example 3.5.7 reveals, the smallest fuzzy vector k(u) of the Mares equivalence class of a fuzzyvector u may not be a Mares core of u, and different skew fuzzy vectors may be Mares equivalent to each other.

2. Representation of fuzzy vectors via support functions

2.1. Convex bodies via support functionsThroughout, let Rn denote the n-dimensional Euclidean space. Following the terminologies of convex geometry

[13, 26], by a convex body in Rn we mean a nonempty, compact and convex subset of Rn; that is, A ⊆ Rn is a convexbody if it is nonempty, closed, bounded and

λs + (1 − λ)t ∈ A

whenever s, t ∈ A and λ ∈ [0, 1]. The set of all convex bodies in Rn is denoted by Cn.Let 〈−,−〉 denote the standard Euclidean inner product of Rn. A hyperplane H in Rn is usually denoted by

H = {t ∈ Rn | 〈t, x0〉 = α} (2.i)

for some x0 ∈ Rn \ {o} and α ∈ R, where o is the origin of Rn, and x0 is called a normal vector of H. Each hyperplaneH given by (2.i) divides Rn into two closed halfspaces

H− := {t ∈ Rn | 〈t, x0〉 ≤ α} and H+ := {t ∈ Rn | 〈t, x0〉 ≥ α}. (2.ii)

Let A ⊆ Rn be closed and convex. For every x ∈ Rn, there exists a unique point pA(x) ∈ A such that

||x − pA(x)|| = d(x, A) :=∧a∈A

||x − a||, (2.iii)

where ||-|| refers to the standard Euclidean norm on Rn. Indeed, the existence of pA(x) is obvious by the closedness of

A. For the uniqueness of pA(x), suppose that qA(x) ∈ A also satisfies (2.iii), but pA(x) , qA(x). ThenpA(x) + qA(x)

2∈ A

by the convexity of A, but∣∣∣∣∣∣∣∣∣∣x − pA(x) + qA(x)2

∣∣∣∣∣∣∣∣∣∣ =

∣∣∣∣∣∣∣∣∣∣ x − pA(x)2

+x − qA(x)

2

∣∣∣∣∣∣∣∣∣∣ < ∣∣∣∣∣∣∣∣∣∣ x − pA(x)2

∣∣∣∣∣∣∣∣∣∣ +

∣∣∣∣∣∣∣∣∣∣ x − qA(x)2

∣∣∣∣∣∣∣∣∣∣ = ||x − pA(x)|| = ||x − qA(x)||;

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that is,pA(x) + qA(x)

2is strictly closer to x than pA(x) and qA(x), contradicting to the fact that pA(x) and qA(x) both

satisfy Equation (2.iii). Thus we obtain a well-defined map

pA : Rn → A,

called the metric projection of A. It is obvious that pA(a) = a for all a ∈ A.

Lemma 2.1.1. (See [13, 26].) If A ⊆ Rn is closed and convex, then the metric projection pA : Rn → A is non-expansive in the sense that

||pA(x) − pA(y)|| ≤ ||x − y||

for all x, y ∈ Rn.

Proof. We only prove the case of x, y ∈ Rn \ A, while the rest cases can be treated analogously. Since the conclusionholds trivially when pA(x) = pA(y), suppose that pA(x) , pA(y). In this case, the convexity of A guarantees that theline segment [pA(x), pA(y)] ⊆ A. Considering the hyperplane

Hx := {t ∈ Rn | 〈t, pA(x) − pA(y)〉 = 〈pA(x), pA(x) − pA(y)〉},

it is clear that pA(y) ∈ H−x . We claim that x ∈ H+x . In fact, if x ∈ Rn \ H+

x , then 〈x − pA(x), pA(y) − pA(x)〉 > 0, andconsequently the angle between x − pA(x) and pA(y) − pA(x) is acute. This means that the line segment [pA(x), pA(y)]contains a point which is strictly closer to x than pA(x), contradicting to the definition of pA(x) (see Equation (2.iii)).Similarly, for the hyperplane

Hy := {t ∈ Rn | 〈t, pA(x) − pA(y)〉 = 〈pA(y), pA(x) − pA(y)〉}

we may deduce that pA(x) ∈ H+y and y ∈ H−y .

Since pA(x) ∈ Hx, pA(y) ∈ Hy and pA(x) − pA(y) is the normal vector of both the hyperplanes Hx and Hy, thedistance between Hx and Hy is precisely ||pA(x) − pA(y)||. From x ∈ H+

x , pA(y) ∈ H−x , pA(x) ∈ H+y and y ∈ H−y we see

that the distance between x and y is no less than the distance between Hx and Hy; that is, ||pA(x)− pA(y)|| ≤ ||x−y||.

Let A ⊆ Rn be a subset, and let H ⊆ Rn be a hyperplane. We say that H supports A at t0 if t0 ∈ A ∩ H and eitherA ⊆ H+ or A ⊆ H−; in this case, t0 necessarily lies in the boundary bd A of A, and H is called a support hyperplane ofA. If a hyperplane H given by (2.i) supports A and A ⊆ H−, then x0 is called an exterior normal vector of H.

Lemma 2.1.2. If A ⊆ Rn is closed and convex, then for each x ∈ Rn \ A, there exists a hyperplane H that supports Aat pA(x), with x − pA(x) being its exterior normal vector.

Proof. LetH = {t ∈ Rn | 〈t, x − pA(x)〉 = 〈pA(x), x − pA(x)〉}.

Then pA(x) ∈ A ∩ H, and we claim that A ⊆ H−. In fact, if there exists z ∈ A ∩ (Rn \ H−), then the line segment[pA(x), z] ⊆ A by the convexity of A. Note that z ∈ Rn \ H− means that 〈z − pA(x), x − pA(x)〉 > 0, and consequentlythe angle between z − pA(x) and x − pA(x) is acute. Thus, the line segment [pA(x), z] must contain a point which isstrictly closer to x than pA(x), contradicting to the definition of pA(x).

The following theorem is well known in convex geometry, and we present a proof here for the sake of self-containment:

Theorem 2.1.3. (See [13, 26].) If A ⊆ Rn is closed and convex, then for each t0 ∈ bd A, there exists a (not necessarilyunique) hyperplane H that supports A at t0.

Proof. Let {tk} ⊆ Rn \ A be a sequence that converges to t0 ∈ bd A, which induces a sequence {pA(tk)} ⊆ bd A. Foreach positive integer k, by Lemma 2.1.2 we may find a hyperplane

Hk = {t ∈ Rn | 〈t, tk − pA(tk)〉 = 〈pA(tk), tk − pA(tk)〉}

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that supports A at pA(tk), with tk − pA(tk) being its exterior normal vector. Let

xk :=tk − pA(tk)||tk − pA(tk)||

.

Then xk belongs to S n−1, the unit sphere in Rn. By the compactness of S n−1, the sequence {xk} has a convergentsubsequence, and without loss generality we may suppose that {xk} itself converges to x0 ∈ S n−1. Note that

||pA(tk) − t0|| = ||pA(tk) − pA(t0)|| ≤ ||tk − t0||

by Lemma 2.1.1, which necessarily forces limk→∞

pA(tk) = t0 as we already have limk→∞

tk = t0. We claim that the hyper-

planeH = {t ∈ Rn | 〈t, x0〉 = 〈t0, x0〉}

supports A at t0, with x0 being its exterior normal vector. To see this, note that t0 ∈ A ∩ H is obvious, and for everypositive integer k we have A ⊆ H−k , which implies that

〈a, tk − pA(tk)〉 ≤ 〈pA(tk), tk − pA(tk)〉, i.e., 〈a, xk〉 ≤ 〈pA(tk), xk〉

for all a ∈ A. By letting k → ∞ in the above inequality we immediately obtain that 〈a, x0〉 ≤ 〈t0, x0〉 for all a ∈ A, andconsequently A ⊆ H−, which completes the proof.

Recall that the support function [13, 26] of a convex body A ∈ Cn is given by

hA : S n−1 → R, hA(x) :=∨a∈A

〈a, x〉, (2.iv)

where S n−1 is the unit sphere in Rn. Obviously, hA is bounded on S n−1; indeed,

|hA(x)| ≤∨a∈A

||a||

for all x ∈ S n−1.

Remark 2.1.4. The domain of the support function of a convex body A ∈ Cn is defined as Rn in [13, 26]; that is,

hA : Rn → R, hA(x) :=∨a∈A

〈a, x〉. (2.v)

In fact, the function hA given by (2.v) is completely determined by its values on S n−1, since it always holds that

hA(o) = 0 and hA(x) = ||x|| · hA

(x||x||

)for all x ∈ Rn \ {o}. Therefore, it does no harm to restrict the domain of hA to S n−1.

Conversely, to each function h : S n−1 → R we may associate a subset

Ah := {t ∈ Rn | ∀x ∈ S n−1 : 〈t, x〉 ≤ h(x)} (2.vi)

of Rn. Convex bodies can be fully characterized through support functions as follows, which is a modification of [13,Theorem 4.3] and [26, Theorem 1.7.1]:

Theorem 2.1.5. A function h : S n−1 → R is the support function of a (unique) convex body Ah if, and only if, h issublinear in the sense that

h(x) + h(−x) ≥ 0

and

h(λx + (1 − λ)y||λx + (1 − λ)y||

)≤λh(x) + (1 − λ)h(y)||λx + (1 − λ)y||

for all x, y ∈ S n−1, λ ∈ [0, 1] with λx + (1 − λ)y , 0.

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Before proving this theorem, let us recall that a function f : Rn → R is convex if

f (λx + (1 − λ)y) ≤ λ f (x) + (1 − λ) f (y)

for all x, y ∈ Rn and λ ∈ [0, 1].

Lemma 2.1.6. (See [13, 26].) A function f : Rn → R is convex if, and only if, its epigraph

epi f := {(x, α) ∈ Rn × R | f (x) ≤ α} ⊆ Rn+1

is convex. In this case, f is necessarily continuous on Rn and, consequently, epi f is closed.

Proof. Step 1. f is convex if, and only if, epi f is convex. If f is convex, for any (x, α), (y, β) ∈ epi f and λ ∈ [0, 1]we have

f (λx + (1 − λ)y) ≤ λ f (x) + (1 − λ) f (y) ≤ λα + (1 − λ)β;

that is, λ(x, α) + (1 − λ)(y, β) = (λx + (1 − λ)y, λα + (1 − λ)β) ∈ epi f . Thus epi f is convex.Conversely, if epi f is convex, for any x, y ∈ Rn and λ ∈ [0, 1] we have

(λx + (1 − λ)y, λ f (x) + (1 − λ) f (y)) = λ(x, f (x)) + (1 − λ)(y, f (y)) ∈ epi f

because (x, f (x)), (y, f (y)) ∈ epi f ; that is, f (λx + (1 − λ)y) ≤ λ f (x) + (1 − λ) f (y), showing that f is convex.Step 2. If f : Rn → R is convex, then f is continuous on Rn. To this end, for any x0 ∈ Rn we choose a simplex

S =

n+1∑i=1

λixi

∣∣∣∣ n+1∑i=1

λi = 1 and λi ≥ 0 for all i = 1, . . . , n + 1

with vertices x1, . . . , xn+1 ∈ Rn, such that there exists an open ball B(x0, ρ) ⊆ S (ρ > 0). Note that f is clearly boundedon S since, by Jensen’s inequality (cf. [26, Remark 1.5.1]),

f (x) = f

n+1∑i=1

λixi

≤ n+1∑i=1

λi f (xi) ≤ c := max{ f (x1), . . . , f (xn+1)}

for all x =

n+1∑i=1

λixi ∈ S . Now, for any y = x0 + λt ∈ B(x0, ρ) (λ ∈ [0, 1], ||t|| = ρ), the convexity of f implies that

f (y) = f (x0 + λt) = f ((1 − λ)x0 + λ(x0 + t)) ≤ (1 − λ) f (x0) + λ f (x0 + t),

and consequently f (y) − f (x0) ≤ λ( f (x0 + t) − f (x0)) ≤ λ(c − f (x0)) because x0 + t ∈ S . Similarly,

f (x0) = f(

11 + λ

y +λ

1 + λ(x0 − t)

)≤

11 + λ

f (y) +λ

1 + λf (x0 − t),

and consequently f (x0) − f (y) ≤ λ( f (x0 − t) − f (x0)) ≤ λ(c − f (x0)). It follows that

| f (y) − f (x0)| ≤ λ(c − f (x0)) =c − f (x0)

ρ||y − x0||

for all y ∈ B(x0, ρ), which immediately implies the continuity of f at x0.Step 3. If f : Rn → R is continuous, then epi f is closed. This is easy since, for any sequence {(xk, αk)} ⊆ epi f

that converges to (x, α), f (x) ≤ α becomes an immediate consequence of f (xk) ≤ αk for all positive integers k inconjunction with the continuity of f , which means precisely that (x, α) ∈ epi f . This completes the proof.

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Proof of Theorem 2.1.5. Necessity. Let A ∈ Cn be a convex body. Then it is clear that

hA(x) + hA(−x) =∨a∈A

〈a, x〉 +∨a∈A

〈a,−x〉 =∨a∈A

〈a, x〉 −∧a∈A

〈a, x〉 ≥ 0

and

hA

(λx + (1 − λ)y||λx + (1 − λ)y||

)=

∨a∈A

⟨a,

λx + (1 − λ)y||λx + (1 − λ)y||

⟩≤

λ∨a∈A

〈a, x〉 + (1 − λ)∨a∈A

〈a, y〉

||λx + (1 − λ)y||=λhA(x) + (1 − λ)hA(y)||λx + (1 − λ)y||

for all x, y ∈ S n−1, λ ∈ [0, 1] with λx + (1 − λ)y , 0.Sufficiency. Let h : S n−1 → R be a sublinear function. Since

Ah = {t ∈ Rn | ∀x ∈ S n−1 : 〈t, x〉 ≤ h(x)} =⋂

x∈S n−1

{t ∈ Rn | 〈t, x〉 ≤ h(x)} (2.vii)

is the intersection of closed halfspaces which are necessarily convex, it is clearly closed and convex. Moreover,considering the standard basis {e1, . . . , en} of Rn we see that each coordinate of t ∈ Ah is bounded by

α0 := max{|h(ei)|, |h(−ei)| | 1 ≤ i ≤ n},

and thus ||t|| ≤√

nα0 for all t ∈ Ah; that is, Ah is bounded. Next, we show that Ah , ∅ and hAh = h, so that Ah is aconvex body and h is the support function of Ah.

Firstly, Ah , ∅. To see this, let us extend h : S n−1 → R to h : Rn → R as elaborated in Remark 2.1.4, i.e.,

h(x) =

0 if x = o,

||x|| · h(

x||x||

)else

for all x ∈ Rn. Thenh(λx) = λh(x) and h(x + y) ≤ h(x) + h(y) (2.viii)

for all x, y ∈ Rn and λ ≥ 0, where the first equality follows obviously from the definition of h, while the secondinequality holds because

h(o) = 0 ≤ ||x|| ·(h(

x||x||

)+ h

(−

x||x||

))= h(x) + h(−x)

for all x ∈ Rn \ {o}, and

h(x + y) = ||x + y|| · h(

x + y||x + y||

)= ||x + y|| · h

||x||

||x|| + ||y||·

x||x||

+||y||

||x|| + ||y||·

y||y||∣∣∣∣∣∣∣∣∣∣ ||x||

||x|| + ||y||·

x||x||

+||y||

||x|| + ||y||·

y||y||

∣∣∣∣∣∣∣∣∣∣

≤ ||x + y|| ·

||x||||x|| + ||y||

· h(

x||x||

)+

||y||||x|| + ||y||

· h(

y||y||

)∣∣∣∣∣∣∣∣∣∣ ||x||||x|| + ||y||

·x||x||

+||y||

||x|| + ||y||·

y||y||

∣∣∣∣∣∣∣∣∣∣ = h(x) + h(y)

for all x, y ∈ Rn with x + y , o.Since h : Rn → R is clearly convex by (2.viii), Lemma 2.1.6 tells us that

epi h = {(x, α) ∈ Rn × R | h(x) ≤ α} ⊆ Rn+1

is closed and convex. By Theorem 2.1.3, there exists a hyperplane Hy that supports epi h at any (y, h(y)) ∈ bd epi hwith y , o. We claim that Hy passes through the origin (o, 0) of Rn ×R = Rn+1, so that Hy also supports epi h at (o, 0).

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In fact, if (o, 0) < Hy, then the straight line passing through (o, 0) and (y, h(y)) intersects the hyperplane Hy in exactlyone point, i.e., (y, h(y)). It follows that (o, 0) and (2y, 2h(y)) are on different sides of Hy. However, it is clear that (o, 0)and (2y, 2h(y)) are both in epi h, which contradicts to the fact that Hy is a support hyperplane of epi h.

Note that the definition of epi h indicates that the exterior normal vector of Hy can be written as the form (ty,−1),which intuitively means that the exterior normal vector of Hy “points below Rn” in Rn × R. Since Hy also supportsepi h at (o, 0), we obtain

Hy = {(x, α) ∈ Rn × R | 〈(x, α), (ty,−1)〉 = 〈(o, 0), (ty,−1)〉 = 0}. (2.ix)

Since epi h ⊆ H−y , for any x ∈ Rn, considering (x, h(x)) ∈ epi h we have

〈ty, x〉 − h(x) = 〈(x, h(x)), (ty,−1)〉 ≤ 0,

i.e., 〈ty, x〉 ≤ h(x). In particular, this means that 〈ty, x〉 ≤ h(x) for all x ∈ S n−1, and consequently ty ∈ Ah (cf. (2.vii)),showing that Ah , ∅.

Secondly, hAh = h. On one hand,hAh (x) =

∨a∈Ah

〈a, x〉 ≤ h(x)

for all x ∈ S n−1 is an immediate consequence of (2.iv) and (2.vii). On the other hand, for any y ∈ S n−1, from(y, h(y)) ∈ Hy we have 〈ty, y〉 = h(y) by (2.ix). Since ty ∈ Ah, it follows that

hAh (y) =∨a∈Ah

〈a, y〉 ≥ 〈ty, y〉 = h(y)

for all y ∈ S n−1. Hence hAh = h.Finally, for the uniqueness of Ah, it remains to show that AhA = A for any convex body A ∈ Cn. Note that A ⊆ AhA

follows easily from (2.iv) and (2.vii). For the reverse inclusion, we proceed by contradiction. Suppose that t0 ∈ AhA

but t0 < A. Since A is closed and convex, by Lemma 2.1.2 we may find a hyperplane H that supports A at pA(t0), with

x0 :=t0 − pA(t0)||t0 − pA(t0)||

∈ S n−1

being its exterior normal vector, i.e.,

H = {t ∈ Rn | 〈t, x0〉 = 〈pA(t0), x0〉}.

Now, it follows from A ⊆ H− that 〈a, x0〉 ≤ 〈pA(t0), x0〉 for all a ∈ A, but t0 ∈ Rn \ H− forces 〈t0, x0〉 > 〈pA(t0), x0〉;that is,

〈t0, x0〉 > 〈pA(t0), x0〉 ≥∨a∈A

〈a, x0〉 = hA(x0),

contradicting to t0 ∈ AhA . This completes the proof.

As an immediate consequence of Theorem 2.1.5, the following characterization of elements of convex bodies willbe useful later (cf. [31, Theorem 2.2]):

Proposition 2.1.7. Let A ∈ Cn be a convex body. Then t ∈ A if, and only if, 〈t, x〉 ≤ hA(x) for all x ∈ S n−1.

In particular, constant support functions correspond to closed balls centered at the origin:

Proposition 2.1.8. A convex body A ∈ Cn is a closed ball centered at the origin if, and only if, its support functionhA : S n−1 → R is a (nonnegative) constant function. In this case, hA is necessarily constant at the radius of A.

8

Proof. Suppose that A is a closed ball of radius λ centered at the origin. Then

hA(x) =∨a∈A

〈a, x〉 = 〈λx, x〉 = λ (2.x)

for all x ∈ S n−1. Conversely, if hA(x) = λ for all x ∈ S n−1, then it follows from Proposition 2.1.7 that

t ∈ A ⇐⇒ ∀x ∈ S n−1 : 〈t, x〉 ≤ λ ⇐⇒ ||t|| =⟨t,

t||t||

⟩≤ λ,

which also guarantees the nonnegativity of λ.

Recall that the Minkowski sum [26] of convex bodies B,C ∈ Cn is given by

B + C := {b + c | b ∈ B, c ∈ C}.

Note that a direct computation

hB+C(x) =∨

b∈B,c∈C

〈b + c, x〉 =∨

b∈B,c∈C

(〈b, x〉 + 〈c, x〉) =∨b∈B

〈b, x〉 +∨c∈C

〈c, x〉 = hB(x) + hC(x)

for any x ∈ S n−1 shows that hB+C = hB + hC (cf. [26, Theorem 1.7.5]), in combination with Theorem 2.1.5 we obtain:

Proposition 2.1.9. For convex bodies A, B,C ∈ Cn, A = B + C if, and only if, hA = hB + hC .

2.2. Fuzzy vectors via convex bodies and support functionsFollowing the terminology of [19], by a fuzzy vector we mean a fuzzy subset of Rn, i.e., a function

u : Rn → [0, 1],

subject to the following requirements:

(V1) u is regular, i.e., there exists t0 ∈ Rn with u(t0) = 1;

(V2) u is compactly supported, i.e., the closure of {t ∈ Rn | u(t) > 0} is compact;

(V3) u is quasi-concave, i.e., u(s) ∧ u(t) ≤ u(λs + (1 − λ)t) for all s, t ∈ Rn and λ ∈ [0, 1];

(V4) u is upper semi-continuous, i.e., {t ∈ Rn | u(t) ≥ α} is closed for all α ∈ [0, 1].

The set of all fuzzy vectors of dimension n is denoted by Fn, and a canonical embedding of Rn into Fn assigns toeach a ∈ Rn a “crisp” fuzzy vector

a : Rn → [0, 1], a(t) :=

1 if t = a,0 else.

Remark 2.2.1. The conditions (V1)–(V4), first appeared in [16] and [24], are originated from the definition of fuzzynumbers, i.e., fuzzy vectors of dimension 1 (see [6, 8, 9, 10, 12]); so, fuzzy vectors are also called n-dimensional fuzzynumbers (see [31, 32, 28, 27]). It should be reminded that n-dimensional fuzzy vectors defined in [27] are differentfrom our fuzzy vectors here.

For each u ∈ Fn and α ∈ [0, 1], the α-level sets of u are defined as

uα :=

{t ∈ Rn | u(t) ≥ α} if α ∈ (0, 1],⋃

α∈(0,1]uα = {t ∈ Rn | u(t) > 0} if α = 0.

It is easy to see thatu(t) =

∨t∈uα

α (2.xi)

for each u ∈ Fn and t ∈ Rn.

9

Remark 2.2.2. Since α ranges in the closed interval [0, 1], the supremum in Equation (2.xi) is computed in [0, 1].Hence, if t < uα for all α ∈ [0, 1], i.e.,

{α | t ∈ uα} = ∅,

then u(t) = 0 because 0, as the bottom element of [0, 1], is the supremum of the empty subset of [0, 1].

It is well known that fuzzy vectors can be characterized via convex bodies as follows, whose proof is straightfor-ward and will be omitted here:

Theorem 2.2.3. (See [22, 17].) Let {Aα | α ∈ [0, 1]} be a family of subsets of Rn. Then there exists a (unique) fuzzyvector

u : Rn → [0, 1], u(t) =∨t∈Aα

α

such thatuα = Aα

for all α ∈ [0, 1] if, and only if,

(L1) Aα is a convex body in Rn for each α ∈ [0, 1];

(L2) Aα ⊇ Aβ whenever 0 ≤ α < β ≤ 1;

(L3) Aα0 =⋂k≥1

Aαk for each increasing sequence {αk} ⊆ [0, 1] that converges to α0 > 0;

(L4) A0 =⋃

α∈(0,1]Aα.

Since all the level sets of a fuzzy vector are convex bodies, it makes sense to define the support function [24, 3] ofu ∈ Fn as

hu : [0, 1] × S n−1 → R, hu(α, x) :=∨t∈uα

〈t, x〉;

that is, hu(α,−) := huα is the support function of the convex body uα for each α ∈ [0, 1]. In particular, hu is boundedon [0, 1] × S n−1, because

hu(α, x) ≤∨t∈u0

〈t, x〉 = hu0 (x)

for all α ∈ [0, 1], x ∈ S n−1, and hu0 is bounded on S n−1.With Theorems 2.1.5 and 2.2.3 we may describe fuzzy vectors through support functions as the following repre-

sentation theorem reveals, which is the main result of this paper:

Theorem 2.2.4. A function h : [0, 1] × S n−1 → R is the support function of a (unique) fuzzy vector

u : Rn → [0, 1]

given byu(t) =

∨t∈Ah(α,−)

α =∨{α | ∀x ∈ S n−1 : 〈t, x〉 ≤ h(α, x)}

if, and only if,

(VS1) h(α,−) : S n−1 → R is sublinear for each α ∈ [0, 1], i.e.,

h(α, x) + h(α,−x) ≥ 0

and

h(α,

λx + (1 − λ)y||λx + (1 − λ)y||

)≤λh(α, x) + (1 − λ)h(α, y)||λx + (1 − λ)y||

for all α ∈ [0, 1], x, y ∈ S n−1, λ ∈ [0, 1] with λx + (1 − λ)y , 0,

10

(VS2) h(−, x) : [0, 1]→ R is non-increasing, left-continuous on (0, 1] and right-continuous at 0 for each x ∈ S n−1.

Proof. Necessity. Let u ∈ Fn be a fuzzy vector. Then hu clearly satisfies (VS1) by Theorem 2.1.5. For (VS2), let usfix x ∈ S n−1. Then hu(−, x) : [0, 1]→ R is non-increasing because of (L2).

To see that hu(−, x) is left-continuous at each α0 ∈ (0, 1], let {αk} ⊆ (0, 1] be an increasing sequence that convergesto α0. Then uα0 =

⋂k≥1

uαk by (L3). For each ε > 0, we claim that there exists a positive integer k such that for all

t ∈ uαk , there exists rt ∈ uα0 with||t − rt || < ε. (2.xii)

Indeed, suppose that we find an ε0 > 0 such that for all positive integers k, there exists tk ∈ uαk such that

d(tk, uα0 ) :=∧

r∈uα0

||tk − r|| ≥ ε0.

Then the sequence {tk} is contained in the compact set uα1 , and thus it has a convergent subsequence. Without loss ofgenerality we may suppose that lim

k→∞tk = t0. Then t0 ∈ uαk for all k ≥ 1 because {tm | m ≥ k} ⊆ uαk , and consequently

t0 ∈⋂k≥1

uαk = uα0 . But the construction of tk forces d(t0, uα0 ) ≥ ε0, which is a contradiction.

Note that (2.xii) actually means thatuαk ⊆ uα0 + Bε ,

where Bε refers to the closed ball of radius ε centered at the origin, since each t ∈ uαk may be written as

t = rt + (t − rt)

with rt ∈ uα0 and t − rt ∈ Bε . Then it follows from Propositions 2.1.8 and 2.1.9 that

hu(αk, x) = huαk(x) ≤ huα0

(x) + hBε (x) = hu(α0, x) + ε.

Hence, together with the monotonicity of hu(−, x) : [0, 1] → R we conclude that limk→∞

hu(αk, x) = hu(α0, x), which

proves the left-continuity of hu(−, x) at α0.To see that hu(−, x) is right-continuous at 0, let ε > 0. The compactness of u0 allows us to find q1, . . . , qk ∈ u0

such that u0 is covered by finitely many open balls

B(q1,

ε

2

), . . . , B

(qk,

ε

2

)centered at q1, . . . , qk, respectively, with radii

ε

2.

Note that for each t ∈ u0 =⋃

α∈(0,1]uα, there exists αt ∈ (0, 1] and st ∈ uαt such that ||t − st || <

ε

2. Let

αq := min{αq1 , . . . , αqk } > 0,

and let B(qt,

ε

2

)(qt ∈ {q1, . . . , qk}) be the open ball containing t. Then sqt ∈ uαqt

⊆ uαq , and

||t − sqt || ≤ ||t − qt || + ||qt − sqt || <ε

2+ε

2= ε.

It follows thatu0 ⊆ uαq + Bε ,

because t = sqt + (t − sqt ) for all t ∈ u0, where sqt ∈ uαq and t − sqt ∈ Bε . By Propositions 2.1.8 and 2.1.9 this meansthat

hu(0, x) = hu0 (x) ≤ huαq(x) + hBε (x) = hu(αq, x) + ε.

Hence, together with the monotonicity of hu(−, x) : [0, 1] → R we conclude that limα→0+

hu(α, x) = hu(0, x), which

proves the right-continuity of hu(−, x) at 0.

11

Sufficiency. It suffices to show that {Ah(α,−) | α ∈ [0, 1]} satisfies the conditions (L1)–(L4).Firstly, (L1) is a direct consequence of Theorem 2.1.5.Secondly, (L2) holds since h(−, x) is non-increasing for all x ∈ S n−1.Thirdly, for (L3), let {αk} ⊆ (0, 1] be an increasing sequence that converges to α0 ∈ (0, 1]. It remains to show that⋂

k≥1

Ah(αk ,−) ⊆ Ah(α0,−),

since the reverse inclusion is trivial by (L2). Suppose that t ∈ Ah(αk ,−) for all k ≥ 1. Then, by Proposition 2.1.7,〈t, x〉 ≤ h(αk, x) for all x ∈ S n−1. Thus the left-continuity of h(−, x) at α0 implies that

〈t, x〉 ≤ limk→∞

h(αk, x) = h(limk→∞

αk, x)

= h(α0, x),

and consequently t ∈ Ah(α0,−).Finally, we prove (L4) by showing that

Ah(0,−) ⊆ A :=⋃

α∈(0,1]

Ah(α,−)

as the reverse inclusion is trivial by (L2). We proceed by contradiction. Suppose that t0 ∈ Ah(0,−) but t0 < A. Notethat A is also a convex body, since A ⊆ Ah(0,−) and A is the closure of the union of a family of convex bodies linearlyordered by inclusion. Thus, by Lemma 2.1.2 we may find a hyperplane H that supports A at pA(t0), with

x0 :=t0 − pA(t0)||t0 − pA(t0)||

∈ S n−1

as its exterior normal vector, i.e.,H = {t ∈ Rn | 〈t, x0〉 = 〈pA(t0), x0〉},

which necessarily satisfies t0 ∈ Rn \ H− and A ⊆ H−. It follows that

h(α, x0) =∨

t∈Ah(α,−)

〈t, x0〉 (Theorem 2.1.5)

≤ 〈pA(t0), x0〉 (Ah(α,−) ⊆ A ⊆ H−)= 〈t0, x0〉 − 〈t0 − pA(t0), x0〉

= 〈t0, x0〉 − ||t0 − pA(t0)||(x0 =

t0 − pA(t0)||t0 − pA(t0)||

)≤ h(0, x0) − ||t0 − pA(t0)|| (t0 ∈ Ah(0,−))

for all α ∈ (0, 1], which contradicts to the right-continuity of h(−, x0) at 0. The proof is thus completed.

3. Mares cores of fuzzy vectors

3.1. Addition of fuzzy vectorsWith the results of Section 2 we are now able to characterize the addition ⊕ of fuzzy vectors through their level

sets and support functions. Explicitly, the sumu ⊕ v ∈ Fn

of fuzzy vectors u, v ∈ Fn is defined by Zadeh’s extension principle (cf. [30, 6]), i.e.,

(u ⊕ v)(t) :=∨

r+s=t

u(r) ∧ v(s)

for all t ∈ Rn.

12

Theorem 3.1.1. For fuzzy vectors u, v,w ∈ Fn, the following statements are equivalent:

(i) u = v ⊕ w.

(ii) uα = vα + wα for all α ∈ [0, 1].

(iii) hu = hv + hw.

Proof. (ii) ⇐⇒ (iii) is an immediate consequence of Proposition 2.1.9. For (i) ⇐⇒ (ii), by Theorem 2.2.3 it sufficesto observe that

(v ⊕ w)α = vα + wα (3.i)

for all α ∈ [0, 1], which is a well-known fact of Zadeh’s extension principle (see, e.g., [23, Proposition 3.3]). For thesake of self-containment we give a proof of (3.i) here. On one hand, vα + wα ⊆ (v ⊕ w)α. Let r ∈ vα, s ∈ wα. Thenv(r) ≥ α and w(s) ≥ α, and consequently

(v ⊕ w)(r + s) ≥ v(r) ∧ w(s) ≥ α,

showing that r + s ∈ (v ⊕ w)α. On the other hand, in order to verify the reverse inclusion, let t ∈ (v ⊕ w)α. Then

(v ⊕ w)(t) =∨

r+s=t

v(r) ∧ w(s) ≥ α.

Consequently, there exists a sequence {rk} in Rn such that

v(rk) ∧ w(t − rk) ≥(1 −

12k

)α ≥

α

2

for any positive integer k. In particular, this means that {rk} ⊆ u α2. Since u α

2is compact, {rk} has a convergent

subsequence, and without loss of generality we may assume that limk→∞

rk = r0. Note that for any positive integers k, l

with l ≥ k,

v(rl) ∧ w(t − rl) ≥(1 −

12l

)α ≥

(1 −

12k

)α.

Letting l→ ∞ in the above inequality, the upper semi-continuity of v,w (see (V4)) then implies that

v(r0) ∧ w(t − r0) ≥(1 −

12k

)α.

Since k is arbitrary, the above inequality forces v(r0) ∧ w(t − r0) ≥ α. It follows that r0 ∈ vα and t − r0 ∈ wα, andtherefore t = r0 + (t − r0) ∈ vα + wα.

3.2. Symmetric fuzzy vectors

Let O(n) denote the orthogonal group of dimension n, i.e., the group of n × n orthogonal matrices.

Definition 3.2.1. A fuzzy vector u ∈ Fn is symmetric (around the origin) if it is O(n)-invariant; that is, if

u(t) = u(Qt)

for all t ∈ Rn and Q ∈ O(n).

We denote by Fns the set of all symmetric fuzzy vectors of dimension n.

Remark 3.2.2. In the case of n = 1, since O(1) = {−1, 1}, u ∈ F1 is symmetric if u(t) = u(−t) for all t ∈ R; that is, uis a symmetric fuzzy number in the sense of Mares [20, 21]. Hence, the symmetry of fuzzy numbers is a special caseof Definition 3.2.1. In fact, as indicated by [25, Remark 2.1], a symmetric fuzzy number actually refers to a fuzzynumber that is symmetric around zero.

13

Theorem 3.2.3. For each fuzzy vector u ∈ Fn, the following statements are equivalent:

(i) u is symmetric.

(ii) For each α ∈ [0, 1], uα is invariant under the action of O(n); that is, Qt ∈ uα for all t ∈ uα and Q ∈ O(n).

(iii) For each α ∈ [0, 1], uα is a closed ball centered at the origin.

(iv) For each α ∈ [0, 1], hu(α,−) : S n−1 → R is a (nonnegative) constant function.

Proof. Since (iii)⇐⇒ (iv) is an immediate consequence of Proposition 2.1.8, it remains to prove that (i) =⇒ (iv) and(iii) =⇒ (ii) =⇒ (i).

(i) =⇒ (iv): Let α ∈ (0, 1] and x ∈ S n−1. For each Q ∈ O(n) and t ∈ uα, the O(n)-invariance of u implies thatQ−1t ∈ uα since u(Q−1t) = u(t) ≥ α, and consequently

〈t,Qx〉 = 〈Q−1t, x〉 ≤ hu(α, x).

Thushu(α,Qx) =

∨t∈uα

〈t,Qx〉 ≤ hu(α, x).

Since Q is arbitrary, it also holds that hu(α,Q−1x) ≤ hu(α, x), and consequently hu(α, x) ≤ hu(α,Qx). Hence

hu(α,Qx) = hu(α, x)

for all Q ∈ O(n). Note that the functionO(n)→ S n−1, Q 7→ Qx

is surjective, and thushu(α, x) = hu(α, y)

for all x, y ∈ S n−1; that is, hu(α,−) : S n−1 → R is constant.In this case, in order to see that the value of h(α,−) is nonnegative, just note that for any t ∈ uα with t , o, from

t||t||∈ S n−1 we deduce that

hu(α,−) = hu

(α,

t||t||

)≥

⟨t,

t||t||

⟩= ||t|| ≥ 0.

(iii) =⇒ (ii): Let α ∈ [0, 1]. If uα is a closed ball of radius λα centered at the origin, then Qt ∈ uα whenever t ∈ uα,since ||t|| ≤ λα obviously implies that ||Qt|| ≤ λα.

(ii) =⇒ (i): Let t ∈ Rn and Q ∈ O(n). If u(t) > 0, then t ∈ uu(t), and consequently Qt ∈ uu(t), i.e., u(Qt) ≥ u(t). AsQ is arbitrary, from u(Q−1t) ≥ u(t) we immediately deduce that u(t) ≥ u(Qt), and thus u(t) = u(Qt).

If u(t) = 0, then t < uα for all α ∈ (0, 1], and consequently Qt < uα for all α ∈ (0, 1], which forces u(Qt) = 0 andcompletes the proof.

From Theorem 3.2.3 we see that the support function hu : [0, 1] × S n−1 → R of a symmetric fuzzy vector u ∈ Fns

actually reduces to a single-variable functionhu : [0, 1]→ R,

and conversely:

Corollary 3.2.4. A function h : [0, 1] → R is the support function of a symmetric fuzzy vector u ∈ Fns if, and only if,

h is nonnegative, non-increasing, left-continuous on (0, 1] and right-continuous at 0.

Proof. Follows immediately from Theorems 2.2.4 and 3.2.3.

As a direct application of Proposition 2.1.8 and Theorem 3.2.3, let us point out the following easy but useful facts:

Corollary 3.2.5. Let A, B ∈ Cn be convex bodies and u, v ∈ Fn be fuzzy vectors.

14

(i) If A and B are both closed balls centered at the origin, then so is A + B, and

λA+B = λA + λB,

where λA, λB and λA+B are the radii of A, B and A + B, respectively.

(ii) If u and v are both symmetric, then so is u ⊕ v.

Proof. For (i), just note that the support function of A + B satisfies

hA+B = hA + hB

by Proposition 2.1.9, and thus hA+B is a (nonnegative) constant function since so are hA and hB. The conclusion thenfollows from Proposition 2.1.8.

For (ii), since u and v are both symmetric, for each α ∈ [0, 1], Theorem 3.2.3 ensures that hu(α,−) and hv(α,−)are both constant on S n−1 which, in conjunction with Theorem 3.1.1, implies that

hu⊕v(α,−) = hu(α,−) + hv(α,−)

is constant on S n−1; that is, u ⊕ v is also symmetric.

3.3. Inner parallel bodies of convex bodiesLet Bλ denote the closed ball of radius λ ≥ 0 centered at the origin. Recall that the inner parallel body (see

[13, 26]) of a convex body A ∈ Cn at distance λ is given by

A−λ = {t ∈ Rn | t + Bλ ⊆ A},

which is also a convex body as long as A−λ , ∅.

Lemma 3.3.1. For convex bodies A, B ∈ Cn and λ ≥ 0,

A = B + Bλ ⇐⇒ hA = hB + λ =⇒ B = A−λ.

Proof. The equivalence of A = B + Bλ and hA = hB + λ follows immediately from Propositions 2.1.8 and 2.1.9. Inthis case, from A = B + Bλ and the definition of A−λ we soon see that B ⊆ A−λ. For the reverse inclusion, suppose thata ∈ A−λ. For each x ∈ S n−1, note that λx ∈ Bλ, and consequently

a + λx ∈ A. (3.ii)

It follows that〈a, x〉 + λ = 〈a, x〉 + 〈λx, x〉 = 〈a + λx, x〉 ≤ hA(x),

where the last inequality is obtained by applying Proposition 2.1.7 to (3.ii). Therefore,

〈a, x〉 ≤ hA(x) − λ = hB(x)

for all x ∈ S n−1, and thus Proposition 2.1.7 guarantees that a ∈ B.

In general, the last implication of Lemma 3.3.1 is proper; that is, B = A−λ does not imply A = B+Bλ. For example,let A and B be the hypercubes of side lengths 4 and 3, respectively, both centered at the origin. Then B = A−1, butA ) B + B1.

We say that a nonempty inner parallel body A−λ of A ∈ Cn is regular if

A = A−λ + Bλ,

or equivalently (see Lemma 3.3.1), ifhA = hA−λ + λ.

For each convex body A ∈ Cn, we write

ΛA := {λ ≥ 0 | A−λ is a regular inner parallel body of A}.

15

Proposition 3.3.2. ΛA is a closed interval, given by

ΛA = [0, λA],

where λA :=∨

ΛA.

Proof. Step 1. A−λA is a regular inner parallel body of A.In order to obtain A−λA + BλA = A, it suffices to show that every t ∈ A lies in A−λA + BλA . Since A = A−λ + Bλ for

all λ ∈ ΛA, for an increasing sequence {λk} ⊆ ΛA that converges to λA we may find ak ∈ A−λk and bk ∈ Bλk with

t = ak + bk

for all positive integers k. Note that both the sequences {ak} and {bk} are bounded, and thus they have convergentsubsequences. Without loss of generality we assume that lim

k→∞ak = a0 and lim

k→∞bk = b0. Then it is clear that

t = a0 + b0

and b0 ∈ BλA , and it remains to prove that a0 ∈ A−λA . To this end, we need to show that a0 + y ∈ A for all y ∈ BλA .Indeed, let {yk} ⊆ BλA be a sequence with lim

k→∞yk = y and yk ∈ Bλk for all positive integers k. Then from ak ∈ A−λk we

deduce that ak + yk ∈ A, and consequently a0 + y ∈ A, as desired.Step 2. If λ ∈ ΛA and 0 ≤ λ′ < λ, then λ′ ∈ ΛA.In order to obtain A−λ′ + Bλ′ = A, it suffices to show that every t ∈ A lies in A−λ′ + Bλ′ . Since A = A−λ + Bλ, we

may find a ∈ A−λ and b ∈ Bλ witht = a + b.

Then it is clear that

t =

[a +

(1 −

λ′

λ

)b]

+λ′

λb

andλ′

λb ∈ Bλ′ , and it remains to prove that a +

(1 −

λ′

λ

)b ∈ A−λ′ . To this end, we need to show that

a +

(1 −

λ′

λ

)b + y′ ∈ A

for all y′ ∈ Bλ′ . Indeed,(1 −

λ′

λ

)b + y′ ∈ Bλ since

∣∣∣∣∣∣∣∣∣∣∣∣(1 −

λ′

λ

)b + y′

∣∣∣∣∣∣∣∣∣∣∣∣ ≤

(1 −

λ′

λ

)||b|| + ||y′|| ≤

(1 −

λ′

λ

)λ + λ′ = λ,

and together with a ∈ A−λ we deduce that a +

(1 −

λ′

λ

)b + y′ ∈ A, which completes the proof.

As an immediate consequence of Proposition 3.3.2, we have the following:

Corollary 3.3.3. For each nonempty subset Λ0 ⊆ ΛA, let λ0 =∨

Λ0. Then A−λ0 is a regular inner parallel body of A,which satisfies

A−λ0 =⋂λ∈Λ0

A−λ and hA−λ0=

∧λ∈Λ0

hA−λ .

Proof. Firstly, with Lemma 3.3.1 we obtain that

hA−λ0= hA − λ0 = hA −

∨Λ0 =

∧λ∈Λ0

(hA − λ) =∧λ∈Λ0

hA−λ .

16

Secondly, if t ∈ A−λ for all λ ∈ Λ0, then Proposition 2.1.7 implies that

〈t, x〉 ≤ hA−λ (x) = hA(x) − λ

for all x ∈ S n−1 and λ ∈ Λ0, and thus〈t, x〉 ≤

∧λ∈Λ0

(hA(x) − λ) = hA−λ0(x)

for all x ∈ S n−1, which means that t ∈ A−λ0 . Hence⋂λ∈Λ0

A−λ ⊆ A−λ0 , which in fact becomes an identity since the

reverse inclusion is trivial.

In particular,A−λA =

⋂λ∈ΛA

A−λ (3.iii)

is the smallest regular inner parallel body of A. We say that a convex body A ∈ Cn is irreducible if

A = A−λA ;

that is, if A does not have any non-trivial regular inner parallel body.Let Cn

i denote the set of irreducible convex bodies in Rn, and let Bn denote the set of closed balls in Rn centered atthe origin. For each convex body A ∈ Cn, the decomposition

A = A−λA + BλA

is unique in the sense of the following:

Theorem 3.3.4. For each convex body A ∈ Cn, there exist a unique B ∈ Cni and a unique Bλ ∈ Bn such that A = B+Bλ.

Moreover, the correspondenceA 7→ (A−λA , BλA )

establishes a bijectionCn ∼←→ Cn

i × Bn,

whose inverse is given by (B, Bλ) 7→ B + Bλ.

3.4. Skew fuzzy vectors and Mares coresMotivated by the notion of skew fuzzy number in the sense of Chai-Zhang [2], we introduce skew fuzzy vectors:

Definition 3.4.1. A fuzzy vector u ∈ Fn is skew if it cannot be written as the sum of a fuzzy vector and a non-trivialsymmetric fuzzy vector; that is, if

u = v ⊕ w

for some v ∈ Fn and w ∈ Fns , then w = o.

Following the terminology from fuzzy numbers [21, 15, 25], Mares cores of fuzzy vectors are defined as follows:

Definition 3.4.2. A fuzzy vector v ∈ Fn is a Mares core of a fuzzy vector u ∈ Fn if v is skew and

u = v ⊕ w

for some symmetric fuzzy vector w ∈ Fns .

These concepts are closely related to inner parallel bodies introduced in Subsection 3.3:

Lemma 3.4.3. For fuzzy vectors u, v ∈ Fn, if u = v ⊕ w for some symmetric fuzzy vector w ∈ Fns , then for each

α ∈ [0, 1],

(i) vα is a regular inner parallel body of uα;

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(ii) hu(α,−) − hv(α,−) : S n−1 → R is a (nonnegative) constant function.

Proof. Since u = v ⊕ w and w is symmetric, Theorem 3.1.1 and Corollary 3.2.4 ensure that

hu(α,−) = hv(α,−) + hw(α), (3.iv)

and thus (ii) holds. For (i), by setting λ = hw(α) and rewriting (3.iv) as huα = hvα + λ it follows soon that vα = (uα)−λand uα = vα + Bλ by Lemma 3.3.1, and hence vα is a regular inner parallel body of uα.

In order to construct a Mares core of each fuzzy vector u ∈ Fn, we start with the following proposition, in which

Υu := {v ∈ Fn | u = v ⊕ w, w ∈ Fns }

is clearly a non-empty set as u ∈ Υu:

Proposition 3.4.4. There is a fuzzy vector c(u) ∈ Fn whose level sets are given by

c(u)α :=

⋂

v∈Υu

vα if α ∈ (0, 1],⋃β∈(0,1]

c(u)β if α = 0,

and whose support function hc(u) : [0, 1] × S n−1 → R is given by

hc(u)(α, x) =

∧

v∈Υu

hv(α, x) if α ∈ (0, 1],

limβ→0+

hc(u)(β, x) if α = 0.

Proof. For the existence of c(u), we show that {c(u)α | α ∈ [0, 1]} satisfies the conditions (L1)–(L3) in Theorem 2.2.3,as (L4) trivially holds.

Firstly, (L2) holds since vα ⊇ vβ for all v ∈ Υu whenever 0 ≤ α < β ≤ 1.Secondly, for (L1), note that for each α ∈ (0, 1], Lemma 3.4.3 tells us that c(u)α is the intersection of a family of

regular inner parallel bodies of uα, and thus c(u)α is itself a regular inner parallel body of uα (see Corollary 3.3.3); inparticular, c(u)α is a convex body. It remains to show that c(u)0 is a convex body. Indeed, c(u)0 is convex since it isthe closure of the union of a family of convex bodies linearly ordered by inclusion, and its boundedness follows fromc(u)0 ⊆ u0.

Thirdly, in order to obtain (L3), let {αk} ⊆ (0, 1] be an increasing sequence that converges to α0 > 0. Thenvα0 =

⋂k≥1

vαk for each v ∈ Υu, and thus

c(u)α0 =⋂v∈Υu

vα0 =⋂v∈Υu

⋂k≥1

vαk =⋂k≥1

⋂v∈Υu

vαk =⋂k≥1

c(u)αk .

For the support function of c(u), let α ∈ (0, 1]. Since c(u)α is the intersection of a family of regular inner parallelbodies of uα, it follows from Corollary 3.3.3 that

hc(u)(α,−) = hc(u)α =∧v∈Υu

hvα =∧v∈Υu

hv(α,−).

Finally, the value of hc(u)(0,−) follows from the right-continuity of hc(u)(−, x) at 0 (see Theorem 2.2.4).

In fact, c(u) also lies in Υu, and it is a Mares core of each fuzzy vector u ∈ Fn:

Theorem 3.4.5. c(u) is skew, and there exists a symmetric fuzzy vector s(u) ∈ Fns such that u = c(u) ⊕ s(u). Hence,

c(u) is a Mares core of u.

18

Proof. Step 1. There exists a symmetric fuzzy vector s(u) ∈ Fns such that u = c(u) ⊕ s(u).

Let h := hu − hc(u) : [0, 1] × S n−1 → R. Note that for each x ∈ S n−1, h(−, x) is left-continuous on (0, 1] andright-continuous at 0 since so are hu(−, x) and hc(u)(−, x) by Theorem 2.2.4. Moreover, as there exists a symmetricfuzzy vector wv ∈ Fn

s such that u = v ⊕ wv for all v ∈ Υu, it follows from Theorem 3.1.1 and Proposition 3.4.4 that

h(α, x) = hu(α, x) − hc(u)(α, x) = hu(α, x) −∧v∈Υu

hv(α, x) =∨v∈Υu

(hu(α, x) − hv(α, x)) =∨v∈Υu

hwv (α)

for all α ∈ (0, 1], x ∈ S n−1. Hence, h is nonnegative, independent of x ∈ S n−1 and non-increasing on α ∈ [0, 1] becauseso is each hwv (v ∈ Υu); that is, h satisfies all the conditions of Corollary 3.2.4. Therefore, h is the support function ofa symmetric fuzzy vector s(u) ∈ Fn

s , which clearly satisfies u = c(u) ⊕ s(u) by Theorem 3.1.1.Step 2. c(u) is skew.Suppose that c(u) = v ⊕ w and w is symmetric. Then hv ≤ hc(u) by Lemma 3.4.3.Conversely, since w and s(u) are both symmetric, so is w ⊕ s(u) by Corollary 3.2.5. Thus, together with

u = c(u) ⊕ s(u) = v ⊕ w ⊕ s(u) = v ⊕ (w ⊕ s(u))

we obtain that v ∈ Υu, which implies that hc(u) ≤ hv by Proposition 3.4.4.Therefore, hc(u) = hv, and it forces w = o, which shows that c(u) is skew.

An obvious application of Theorem 3.4.5 is to determine whether a fuzzy vector is skew:

Corollary 3.4.6. A fuzzy vector u ∈ Fn is skew if, and only if, u = c(u).

Proof. The “if” part is already obtained in Theorem 3.4.5. For the “only if” part, just note that Υu = {u} if u is skew,and thus u = c(u) necessarily follows.

Let Fnk denote the set of skew fuzzy vectors of dimension n. Theorem 3.4.5 actually induces a surjective map as

follows:

Corollary 3.4.7. The assignment (v,w) 7→ v ⊕ w establishes a surjective map

Fnk × F

ns → Fn.

Unfortunately, as the following Example 3.4.9 reveals, unlike Theorem 1.2.1 for the case of n = 1 or Theorem3.3.4 for convex bodies, in general the map given in Corollary 3.4.7 may not be injective. In other words, a fuzzyvector may have many Mares cores, so that there may be many ways to decompose a fuzzy vector as the sum of askew fuzzy vector and a symmetric fuzzy vector!

As a preparation, let us present a sufficient condition for a fuzzy vector to be skew that is easy to verify:

Lemma 3.4.8. Let u ∈ Fn be a fuzzy vector. If the 0-level set u0 of u is an irreducible convex body, then u is skew.

Proof. Suppose that u = v ⊕ w and w is symmetric. Then uα = vα + wα for all α ∈ [0, 1]. In particular, u0 = v0 + w0.Since u0 is irreducible, w0 must be trivial, i.e., w0 = {o}, where o is the origin of Rn. The condition (L2) of Theorem2.2.3 then forces wα = {o} for all α ∈ [0, 1], which means that w = o.

Example 3.4.9. Suppose that n ≥ 2. For each α ∈ [0, 1], let

Aα =

n∏i=1

[α − 1, 1 − α]

be the hypercube in Rn centered at the origin whose edge length is 2 − 2α.For every λ ∈ [0, 1], we may construct a fuzzy vector vλ ∈ Fn whose level sets are given by

(vλ)α = Aα + Bλα,

19

and a non-trivial symmetric fuzzy vector wλ ∈ Fns whose level sets are given by

(wλ)α = B2−λα.

Then there exists a fuzzy vector u ∈ Fn with

uα = Aα + B2 = Aα + Bλα + B2−λα = (vλ)α + (wλ)α

for all λ ∈ [0, 1], where the second equation follows from Corollary 3.2.5. Hence

u = vλ ⊕ wλ

for all λ ∈ [0, 1]. As (vλ)0 = A0 =

n∏i=1

[−1, 1] is irreducible, from Lemma 3.4.8 we know that every vλ is skew, and

therefore every vλ (0 ≤ λ ≤ 1) is a Mares core of u.

With Theorem 1.2.1 and Example 3.4.9 we can now conclude:

Theorem 3.4.10. Every fuzzy vector u ∈ Fn has a unique Mares core if, and only if, the dimension n = 1.

3.5. Mares equivalent fuzzy vectorsThe following definition is also originated from fuzzy numbers (see [21, 25, 2]):

Definition 3.5.1. Fuzzy vectors u, v ∈ Fn are Mares equivalent, denoted by u ∼M v, if there exist symmetric fuzzyvectors w,w′ ∈ Fn

s such thatu ⊕ w = v ⊕ w′.

The relation ∼M is clearly an equivalence relation on Fn, and we denote by [u]M the equivalence class of eachu ∈ Fn.

Proposition 3.5.2. For fuzzy vectors u, v ∈ Fn, the following statements are equivalent:

(i) u ∼M v.

(ii) For each α ∈ [0, 1], either uα is a regular inner parallel body of vα, or vα is a regular inner parallel body of uα.

(iii) For each α ∈ [0, 1], the function hu(α,−) − hv(α,−) is constant on S n−1.

Proof. (ii) ⇐⇒ (iii) is an immediate consequence of Theorem 3.1.1 and Lemma 3.3.1, and (i) =⇒ (iii) follows soonfrom Theorems 3.1.1 and 3.2.3. For (iii) =⇒ (i), let us fix x0 ∈ S n−1, and let η be a common upper bound of hu and hv

on [0, 1] × S n−1. Then the functions[0, 1] → Rα 7→ η + hu(α, x0)

and[0, 1] → Rα 7→ η + hv(α, x0)

clearly satisfy the conditions of Corollary 3.2.4, and thus they are support functions of symmetric fuzzy vectorsw,w′ ∈ Fn

s , respectively.Since hu(α,−) − hv(α,−) is constant on S n−1 for each α ∈ [0, 1], it follows that

hu(α, x) − hv(α, x) = hu(α, x0) − hv(α, x0) = hw(α) − hw′ (α)

for all α ∈ [0, 1], x ∈ S n−1; that is,hu + hw′ = hv + hw,

and therefore u ⊕ w′ = v ⊕ w by Theorem 3.1.1, showing that u ∼M v.

20

Remark 3.5.3. For fuzzy vectors u, v ∈ Fn, u ∼M v does not necessarily imply that u = v ⊕ w or v = u ⊕ w for somew ∈ Fn

s . For example, if

u(t) =

1 if t ∈ B1,

0 if t < B1and v(t) =

1 −||t||2

if t ∈ B2,

0 if t < B2,

then u, v ∈ Fns and it trivially holds that u ⊕ v = v ⊕ u; hence u ∼M v. However, there is no w ∈ Fn

s such that u = v ⊕wor v = u ⊕ w. Indeed, the level sets of u, v are given by

uα ≡ B1 and vα = B2−2α

for all α ∈ [0, 1]; that is, uα is a regular inner parallel body of vα when 0 ≤ α ≤12

, and vα is a regular inner parallel

body of uα when12≤ α ≤ 1.

It is obvious that c(u) ∼M u for all u ∈ Fn. In fact, every Mares core of u is Mares equivalent to u. In what followswe construct another skew fuzzy vector k(u) that is Mares equivalent to u but may not be a Mares core of u:

Proposition 3.5.4. There is a skew fuzzy vector k(u) ∈ Fn whose level sets are given by

k(u)α :=

⋂

v∈[u]M

vα if α ∈ (0, 1],⋃β∈(0,1]

k(u)β if α = 0,

and whose support function hk(u) : [0, 1] × S n−1 → R is given by

hk(u)(α, x) =

∧

v∈[u]M

hv(α, x) if α ∈ (0, 1],

limβ→0+

hk(u)(β, x) if α = 0.

In particular, k(u) ∼M u.

Proof. The verification of k(u) being a fuzzy vector is similar to Proposition 3.4.4 under the help of Proposition 3.5.2,and thus we leave it to the readers. In particular, k(u) ∼M u is an immediate consequence of Proposition 3.5.2 and thefact that each k(u)α is a regular inner parallel body of uα.

To see that k(u) is skew, suppose that k(u) = v ⊕ w and w is symmetric. Then hv ≤ hk(u) by Lemma 3.4.3.Conversely, since k(u) ∼M v, it holds that u ∼M v, and thus hk(u) ≤ hv. Hence hk(u) = hv, which forces w = o andcompletes the proof.

Remark 3.5.5. For each α ∈ (0, 1], the α-level set

uα = {t ∈ Rn | u(t) ≥ α}

of a fuzzy vector u ∈ Fn has a smallest regular inner parallel body given by Equation (3.iii) below Corollary 3.3.3. Itis then tempting to ask whether k(u) could be determined by the α-level sets

k(u)α = (uα)−λuα.

Unfortunately, this is not true since, in general, {(uα)−λuα| α ∈ [0, 1]} does not satisfy the conditions of Theorem 2.2.3

even when n = 1. For example, the level sets of the fuzzy number

u : R→ [0, 1], u(t) =

1 −t2

if t ∈ [0, 2],

0 else

21

are given byuα = [0, 2 − 2α]

for all α ∈ [0, 1], but{(uα)−λuα

| α ∈ [0, 1]} = {{1 − α} | α ∈ [0, 1]}

consists of non-identical singleton sets which obviously violate the condition (L2).

From the definition it is clear that k(u) is the smallest fuzzy vector in the Mares equivalence class of u ∈ Fn; thatis,

hk(u) ≤ hv

for all v ∈ [u]M . In fact, if the dimension n = 1, k(u) is precisely the (unique) Mares core of a fuzzy number uconstructed in [2, Proposition 4.2] (cf. [2, Remark 4.4]); that is:

Corollary 3.5.6. For each fuzzy number u, it holds that

k(u) = c(u).

Moreover, k(u) is the unique Mares core of u and the unique skew fuzzy number in the Mares equivalence class [u]M .

However, if the dimension n ≥ 2, the following continuation of Example 3.4.9 shows that k(u) may not be a Marescore of u, and skew fuzzy vectors may be Mares equivalent to each other:

Example 3.5.7. For the fuzzy vectors u, vλ, wλ (0 ≤ λ ≤ 1) considered in Example 3.4.9, since

vλ ∈ [u]M

for all λ ∈ [0, 1], all skew fuzzy vectors vλ are Mares equivalent to each other. Moreover, although

v0 = k(u) = k(vλ),

v0 is not a Mares core of vλ whenever λ ∈ (0, 1]; in fact, since vλ is skew, its Mares core must be itself.

4. Concluding remarks

The well-known Theorem 2.2.3 builds up a bridge from convex geometry to fuzzy vectors. From the viewpointof a convex geometer, the theory of fuzzy vectors is a theory about sequences of convex bodies. Our main result,the representation of fuzzy vectors via support functions (Theorem 2.2.4), can never be achieved without the aid ofconvex geometry. We believe that the power of convex geometry in the study of fuzzy vectors is still to be unveiled,which is worth further exploration.

We end this paper with two questions about Mares cores of fuzzy vectors that remain unsolved:

(1) Is it possible to find all the Mares cores of a given fuzzy vector?

(2) Is it possible to find a necessary and sufficient condition to characterize those fuzzy vectors with exactly oneMares core?

Acknowledgment

The authors acknowledge the support of National Natural Science Foundation of China (No. 11501393 and No.11701396), and the first named author also acknowledges the support of Sichuan Science and Technology Program(No. 2019YJ0509) and a joint research project of Laurent Mathematics Research Center of Sichuan Normal Universityand V. C. & V. R. Key Lab of Sichuan Province.

The authors are grateful for helpful comments received from Professor Hongliang Lai and Professor BaochengZhu.

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