comparative properties of three metrics in the space of compact convex sets

Set-Valued Analysis 5: 267289, 1997. 267c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Comparative Properties of Three Metricsin the Space of Compact Convex Sets ?

PHIL DIAMOND1,?? PETER KLOEDEN2,z ALEXANDER RUBINOV2 andALEXANDER VLADIMIROV1 zz1Mathematics Department, University of Queensland, 4072 Australia2School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat,Victoria 3353, Australia

(Received: 4 February 1997; in final form: 2 September 1997)

Abstract. Along with the Hausdorff metric, we consider two other metrics on the space of convexsets, namely, the metric induced by the Demyanov difference of convex sets and the BartelsPallaschke metric. We describe the hierarchy of these three metrics and of the corresponding normsin the space of differences of sublinear functions. The completeness of corresponding metric spacesis demonstrated. Conditions of differentiability of convex-valued maps of one variable with respectto these metrics are proved for some special cases. Applications to the theory of convex fuzzy setsare given.

Mathematics Subject Classifications (1991). Primary 52A20; secondary 47D20, 58C06, 90C25.Key words: space of convex sets, Demyanov difference, BartelsPallaschke metric, derivative ofset-valued function, convex fuzzy set.

1. Introduction

The notion of difference of convex sets is extensively used in nonsmooth opti-mization theory, [2, 3, 8, 11]. The difference of two convex sets is usuallyassociated with the arithmetic difference of their support functions. The latter,however, is not necessarily a support function of a convex set. As is known, theclass of support functions to convex sets in Rn coincides with the class of allsublinear functions in Rn, that is of all positively homogeneous convex func-tions. The subtraction operation on support functions of convex sets determinesan equivalence relation on the pairs of convex sets, namely, (A;B) (C;D) iffA+D = B + C , where A+D is the Minkowski sum.

? This research has been supported by the Australian Research Council Grant A 49330974.?? Author for correspondence.z Permanent address of Kloeden: Fachbereich Mathematik, Iohann Wolfgang Goethe Universi-

taet, D-60054, Frankfurt am Main, Germany.zz Permanent address of Vladimirov: Institute of Information Transmission Problems Russian

Academy of Science, Moscow, Russia.

VTEX (Ju) PIPS No.: 149865 MATHKAPSVAN329.tex; 10/11/1997; 12:47; v.7; p.1

268 PHIL DIAMOND ET AL.

Any appropriate definition of a difference A B of two convex sets shouldproduce the same result for all equivalent pairs, that is

AB = C D if (A;B) (C;D):Among definitions of the difference operation on convex sets, the Demyanovdifference looks the most convenient for optimization purposes as is shown, forexample, in [8]. Note that, unlike the well known HukuharaPontryagin dif-ference [12, 19], the Demyanov difference is nonempty for an arbitrary pairof convex compact sets. A metric on the set Y of nonempty convex compactsets in Rn arises as the weakest metric such that the Demyanov difference iscontinuous from Y Y with the metric to Y with the Hausdorff metric; letus call this the Demyanov metric or D-metric. Another related metric is theBartelsPallaschke metric or BP-metric [5].

In this paper we compare the three metrics on the space Y and the associatednorms on the linear space of equivalent pairs of convex sets. We find someinteresting properties of D- and BP-metrics which can make them useful both innon-smooth analysis and fuzzy control theory.

A problem of special interest for us is the properties of convex-valued func-tions, that is maps with compact convex values, in particular, their continuityand differentiability, see, for example, [3, 4, 6, 14, 17, 29]. The hierarchy of H-,D- and BP-metrics on Y revealed in this paper makes it possible to use the mostappropriate definition of the derivative of a convex-valued function according tothe particular problem in non-smooth analysis, fuzzy control, or elsewhere.

2. Preliminaries

Denote by Y the set of all nonempty convex compact sets in Rn. The supportfunction pA of a set A 2 Y is defined on Rn by the formula

pA(x) = maxv2Ahv; xi:

It is well known (see, for example [8]) that the mapping A 7! pA called theMinkowski duality is one-to-one correspondence between Y = Y (Rn) and theset P = P (Rn) of all finite sublinear functions defined on Rn. The Minkowskiduality is known to preserve algebraic operations and the order relation. Wesuppose that the usual algebraic operations together with the order relation areintroduced in Y , namely the Minkowski addition:A+B = fa+b: a 2 A; b 2 Bgand multiplication by nonnegative number A = fa: a 2 Ag, and also the orderrelation by inclusion: A > B () A B.

Recall the definition of the Hausdorff metric H(; ) on Y :H(A;B) = maxfe(A;B); e(B;A)g;

wheree(A;B) = sup

x2Ainfy2Bkx yk:

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COMPARATIVE PROPERTIES OF THREE METRICS 269

It is well-known that the Minkowski duality associates the Hausdorff metric withthe uniform metric on the unit sphere Sn1 = fx 2 Rn: kxk = 1g:

H(A;B) = maxx2Sn1

jpA(x) pB(x)j;

where pA and pB are the support functions of the convex sets A and B. Thenorm of the set A is defined as H(A; 0):

kAk = supx2Akxk:

Clearly kAk = maxkxk=1 jpA(x)j.For any h 2 Rn and A 2 Y , we denote

A(h) =nx 2 A: hh; xi = max

y2Ahh; yi

o;

that is A(h) is a face of A.Let A 2 Y . We shall denote by TA the set of all points v 2 Rn such that

hv; xi attains its maximum on x 2 A at a unique point A(v). The set TA isof full measure in Rn. It is well-known that v 2 TA if and only if there existsthe gradient rpA(v) of the support function pA at the point v. In this caseA(v) = rpA(v):

An ordered pair (A;B) of elements of Y is said to be equivalent to anothersuch pair (C;D) if A+D = B+C. The equivalence of two pairs will be denotedby (A;B) (C;D). The following definition was introduced in [22]:

DEFINITION 2.1. The binary operation defined on Y is called a differenceof convex sets if

(i) (A;B) (C;D) implies AB = C D;(ii) A f0g = A.

It is easy to check (see [22]) that the equality A = B + C implies C = AB.Thus, for example, the Minkowski difference A B = A + (B) is not adifference in this sense because AA = f0g only if A is a singleton.

3. The Demyanov Difference

A special difference of convex sets was introduced in [23] as follows. Let A;B 2Y and let T be a subset of TA \ TB of full measure. Define

A . B = cl cofA(v) B(v): v 2 Tg;where clA is the closure of A and coA is the convex hull of A. This defini-tion does not depend on the choice of the set T , see [8]. Conditions (i) and (ii)apparently hold for the operation ., which we will call the Demyanov difference

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or the D-difference. This construction was applied implicitly by V. F. Demyanovfor the study of connections between the Clarke subdifferential and the quasidif-ferential [7]. We can also represent the Demyanov difference in terms of supportfunctions:

A . B = cl co frpA(v) rpB(v): v 2 Tg:The following assertion was proved in [22]:

PROPOSITION 3.1. The support function pA .B of Demyanov difference satis-fiespA .B(v) = sup

u2Rn([pA(u+ v) pB(u+ v)] [pA(u) pB(u)]):

Let us list several elementary properties of the Demyanov difference in the fol-lowing

LEMMA 3.1.(1) For any pair A;B 2 Y ,

A . B = (B . A):(2) For any nonsingular n n-matrix M and A;B 2 Y ,

MA .MB = M(A . B):(3) If B A then 0 2 A . B.(4) For any pair A;B 2 Y ,

(A . B) +B A:(5) (A . B) = A . B.(6) (A1 +A2) . (B1 +B2) (A1 . B1) + (A2 . B2).

Proof. Properties (3), (5), and (6) have been proved in [22]. Both proper-ties (1) and (2) follow immediately from the definitions. Property (4) followsfrom Proposition 3.1, where u is set to 0. 2

The following example shows that the Demyanov difference is not continuouswith respect to the Hausdorff metric.

EXAMPLE 3.1. Let us consider a family of segments in R2:

S = [(0; 0); (cos; sin)]:

If 6= then there exists a vector v 2 TS \ TS such that S(v) = 0 andS (v) = (cos ; sin ). Thus kS . S k = 1 if 6= and S . S = f0g if = . The Hausdorff distance H(S; S ), however, tends to zero as j j! 0.

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The following assertion can be used for an inductive definition of the D-differencewhich does not use sets of full measure.

LEMMA 3.2. For any A;B 2 Y ,

A . B = cl co[

v2Sn1(A(v) . B(v)): (3.1)

Proof. We only have to prove thatA(v) . B(v) A . B (3.2)

for any v 2 Sn1. This will be proven if we show that, for any x 2 T (A(v))and any " > 0 there exists a y 2 T (A) such that kx yk < ". Indeed, anyx 2 T (A(v)) is an extreme point of A, that is a point which does not belong tothe relative interior of any segment [x; y] A. As is known ([28], see also [20]),the set of exposed points of A, that is T (A), is dense in the set of its extremepoints. Let us prove this fact for completeness.

Suppose x 2 A is an exposed point. For some " > 0 consider the setAnB(x; ") and its closed convex hull A". The point x does not belong to A" forany " > 0, otherwise there would exist a sequence of segments [yi0; yi1] in A suchthat kyi1 yi0k = 2", (yi1 yi0)=2! x as i!1 and, hence, x = (y1 y0)=2 forsome y0; y1 2 A, ky1 y0k = 2". Therefore x can be strongly separated fromA" by a vector v":

hv"; xi > maxy2A"hv"; yi: (3.3)

Moreover, (3.3) holds for all v in some open neighborhood of v". Since T (A) isa set of full measure, there is a p 2 T (A) such that

hp; xi > maxy2A"hp; yi: (3.4)

Hence kxA(p)k < ".Therefore

A(v) . B(v) cl (A . B) = A . B: 2

THEOREM 3.1. The D-difference is well defined by the property (3.1) togetherwith the following three properties:(1) For any x 2 Rn,

A . fxg = A x = fy x: y 2 Ag:

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(2) If A L1 and B L2, where L1 = fx: hh; xi = a1g and L2fx: hh; xi =a2g for some h 2 Rn, khk = 1, than

A . B = (a1 a2)h+ (A0 . B0); (3.5)where A0 and B0, respectively, are the orthogonal projections of A and Bon the hyperplane L = fx: hh; xi = 0g.

(3) If both A and B belong to the same hyperplane L = fx 2 Rn: hh; xi = 0gand M is a linear isometry between L and Rn1 then

M(A . B) = (MA) . (MB): (3.6)Proof. Properties (1) and (2) follow from the definition of the D-difference.

Property (3) follows from the fact that neither h nor h belong to T (A)\T (B)unless both A and B are singletons.

Next, the D-difference A . B in Rn is defined via the D-differences betweencorresponding (n 1)-dimensional faces of A and B. Each of the latter, bymeans of (3.1), (3.5), and (3.6) can be reduced to the difference of certain compactconvex sets in Rn1. Finally, the D-difference in R1 is reduced to the D-differencebetween singletons. 2

4. The Demyanov Metric

Let us introduce a metric D on Y such that the Demyanov difference is acontinuous binary operation from YD YD to YH, where by YD and YH wedenote the space Y provided with metrics D and H respectively. Define theD-metric as

D(A;B) = supv2TkA(v) B(v)k; (4.1)

where T is a subset of TA \ TB of full measure. Again, as for the Demyanovdifference, this definition does not depend on T .

Obviously, D(A;B) = 0 iff A = B, otherwise it is positive. The triangleinequality

D(A;C) 6 D(A;B) + D(B;C)

easily follows from the definition if we take T = TA \ TB \ TC . As followsfrom Example 3.1, the space YD is not separable. Indeed, the family of sets S,0 6 < , is uncountable and the distance between any two of its membersis 1.

Since the function v 7! rp(v) is positively homogeneous of degree zero forany function p which is positively homogeneous of degree one, we have

D(A;B) = supv2TA\TB ; kvk=1

krpA(v) rpB(v)k;

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or, equivalently,

D(A;B) = kA . Bk:Thus the Demyanov metric is defined by means of gradients of support functionstaken at the points of the unit sphere Sn1. At the same time the Hausdorffmetric is defined by the C-norms of restrictions of support functions to Sn1.

A statement analogous to Theorem 3.1 holds for the D-metric:

THEOREM 4.1. The Demyanov metric D is well defined by the following threeproperties:

(1) D(fxg; fyg) = kx yk;(2) if L = L0 L?0 and A = A1 A2, B = B1 B2, where A1; B1 2 L0,

A2; B2 2 L?0 , thenD(A;B) =

q(D(A1; A2))2 + (D(B1; B2))2;

(3) D(A;B) = supv2Sn1

D(A(v); B(v)): (4.2)Proof. First, let us prove that the Demyanov metric enjoys properties (1)(3).

Property (1) is obvious. Property (2) follows fromT (A1 A2) = T (A1) T (A2):

Property (3) follows from (3.2).On the other hand, the same way as in the proof of Theorem 3.1, induction

in n shows that there cannot be two different metrics with properties (1)(3). 2

As follows from (4.2), D-convergence of a sequence fAig, Ai 2 Y , to A 2 Yimplies D-convergence of each sequence fAi(v)g, v 2 Sn1, to A(v). Thereverse statement is not true as the following example shows.

EXAMPLE 4.1. Let A be a set in R3 such that

A(h) = [(1; 0; 0); (1; 0; 0)]for h = (0; 0; 1) and A(v) is a singleton for each v 6= h, v 6= 0. For example, Acan be chosen as the union of all balls Bt, 1 6 t 6 1, where

Bt = f(x; y; z):q

(x t)2 + y2 + (z (2 jtj))2 6 2 jtjg:Then, choose a sequence Ai in such a way that each Ai belongs to A, Ai(h) =A(h), and the sets A and Ai are identical outside the 1=i-neighborhood of A(h).It is easy to show that, for each i = 1; 2 : : :, a set Ai can be chosen such thatit has a face Ai(hi), hi 6= h, which is a segment of length 2 close to A(h).Thus, from (4.2), D(Ai; A) > 2, i = 1; 2; : : :, and the sequence Ai does notD-converge to A though, apparently, any sequence of faces Ai(v) D-convergesto A(v).

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Let us compare the Demyanov and the Hausdorff metrics.

LEMMA 4.1. The Demyanov metric majorizes the Hausdorff metric, that isD(A;B) > H(A;B); A;B 2 Y: (4.3)

Proof. For any pair A;B 2 Y we haveH(A;B) = max

v2Sn1jpA(v) pB(v)j;

hence

H(A;B) = jpA(h) pB(h)jfor some h 2 Sn1. As follows from property (2) in Theorem 4.1,

D(A(h); B(h)) > jpA(h) pB(h)j = H(A;B):It remains to use (4.1). 2

As Example 3.1 shows, the D-topology induced by the Demyanov metric isstrictly stronger than the H-topology induced by the Hausdorff metric.

LEMMA 4.2. The Demyanov difference is a continuous binary operation fromYD YD to YH. It is not continuous in the D-metric for n > 1.

Proof. Suppose Ai D! A and Bi D! B as i ! 1. Consider a set T Rndefined as the intersections of all sets T (Ai), T (Bi), T (Ai . Bi), T (A), T (B)and T (A . B). The Lebesgue measure is -additive, hence T is still a subsetof full measure in Rn. For each " > 0 there is a K = K(") such that

kAi(v)A(v)k < "; kBi(v)B(v)k < "for any v 2 T and i > K("). Hence

k(Ai(v)Bi(v)) (A(v) B(v))k < 2"; v 2 T; i > K("):From the definition of the D-difference it follows that

H((Ai. Bi); (A . B)) < 2"; i > K("):

The second part of the lemma follows from Example 5.2. 2

5. The BartelsPallaschke Metric

Denote by Q = Q(Rn) the vector space of differences pq of sublinear functionsp; q on Rn. The following result was proved in [5]:

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THEOREM 5.1. Let 2 Q(Rn). Denote

kk = minp;q2P (Rn);=pq

nmax

nmaxkxk61

p(x); maxkxk61

q(x)oo; (5.1)

where P (Rn) is the set of sublinear functions on Rn. Then Q(Rn) with the normk k is a Banach space.

Definition (5.1) induces a metric on the space P . Using the Minkowski dualitywe can extend this metric to the space Y , let us call it the BartelsPallaschkemetric or BP-metric and denote it by

BP(A;B) = inf(C;D)(A;B)

maxfkCk; kDkg:

The relation between the BP-metric and the D-metric is revealed by the following

THEOREM 5.2. For any pair A;B 2 Y ,D(A;B) 6 2BP(A;B); (5.2)

that is the BartelsPallaschke metric is not weaker than the Demyanov metric.Proof. Suppose A + D = B + C for some C;D 2 Y . Let v 2 Sn1, then

(A+D)(v) = A(v)+D(v) and (B+C)(v) = B(v)+C(v), where, for instance,

(A+D)(v) =nx 2 A+Djhv; xi = max

y2A+Dhv; yi

o:

For any " > 0, there exists a v 2 TA \ TB such thatkA(v) B(v)k > D(A;B) ";

hence A(v) +D(v) = B(v) + C(v) implies

maxfkCk; kDkg > D(A;B)2

: (5.3)

Since (5.3) is true for any pair C;D such that A +D = B + C , the inequality(5.2) follows. 2

As the following example shows, the BartelsPallaschke metric is strictly strongerthan the Demyanov metric.

EXAMPLE 5.1. Let A be the unit circle fx: kxk 6 1g in R2 and Bk be thesequence of regular polygones with k vertices inscribed into A. It is easy tosee that D(Bk; A) ! 0 as k ! 1. Let us show that BP(Bk; A) > 1 for anyk = 1; 2; : : :. Suppose Bk+C = A+D = E. The boundary of E consists exactlyof k segments Jki which are shifted faces of Bk interspersed with correspondingshifts of pieces Dki of boundary of the set C. The total length of the pieces Dki

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equals the total length of the boundary of C. Each boundary point of E is also aboundary point of some ball B(x; 1) E, because E = A+D. Thus the lengthof any piece Dki of boundary of E is not less than 2=k, hence the total lengthof the boundary of C is not less than 2. Therefore kCk > 1 because, amongall convex sets with a given circumference, the circle has the least diameter.

As follows from Example 3.1, the space YBP is not separable. Example 5.2 belowproves the following

PROPOSITION 5.1. The Demyanov difference is not continuous with respect tothe BartelsPallaschke metric.

EXAMPLE 5.2. Let

A = f(x; y) 2 R2: jxj 6 1; jyj 6 1g and B = f(x; x) 2 R2: 0 6 x 6 g:The difference A = A . B is a polygon and the angle of some of its edgesvaries continuously with , see Figure 1. The length of these edges is not lessthan 1 for all , therefore the distance between any two of them for 1 6= 2is not less then 1 either. Hence, according to (4.2), the sequence fAg does notD-converge to A as ! 0, though, apparently, B ! f0g as ! 0.

The next lemma is a counterpart of the representation (4.2).

LEMMA 5.1. For any pair A;B 2 Y and any v 2 Sn1,BP(A;B) > BP(A(v); B(v)) (5.4)

and the inequality in (5.4) can be strict.

-

6

hhhhhhhhhh

EEEEEEEEEEhhhh

hhhhhhEEEEEEEEEE

....................k

JJ]

A

B

A . B

Figure 1. Discontinuity of the D-difference in the D-metric.

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Proof. The inequality follows from a simple fact that (A+B)(v) = A(v) +B(v) and the definition of BP. It is strict for sets A and Bk in Example 5.1. 2

6. Hierarchy of the Three Norms on Q(Rn)Along with the BartelsPallaschke norm k k (5.1) which will be also denot-ed k kBP, we will consider the following norms on Q(Rn) generated by theHausdorff metric and by the Demyanov difference:

kkH = maxkvk=1 j(v)j; (6.1)

kkD = supkvk=1;v2T

jr(v)j; (6.2)

where T is the set of all points of differentiability of the function .

LEMMA 6.1. The Demyanov norm is stronger than the Hausdorff one and weak-er than the BartelsPallaschke one.

Proof. It follows from the same relations between the metrics H, D, andBP. 2

From Theorem 5.1 and the closed graph theorem for Banach spaces, [10], we canconclude that the space Q(Rn) with either the Hausdorff norm or the Demyanovnorm is not complete. It is well known that the completion of the space (Q(Rn),k kH) is isometric to the space C(Sn1) of all continuous functions defined onthe sphere Sn1 with the uniform norm. We can identify this space with the spaceC0(Rn) of continuous positively homogeneous of degree one functions definedon Rn with the norm kk = maxkvk=1 j(v)j.

The norm k kD on Q(Rn) is equivalent to the norm of the Sobolev spaceW 1;1(Sn1), that is the space of functions f on Sn1 such that its gradientrf(x) taken in the sense of distributions, belongs to L1(Sn1). To be moreprecise, each of n2 scalar distributions (rf(x))ij belongs to L1(Sn1). Indeed,the norm of a function in W 1;1 is defined as the sum of the L1-norm ofr and the supremum of jj. The former, however, majorizes the latter forhomogeneous functions (x).

If n > 1, then clQ(Rn) 6= W 1;1(Sn1) as can be seen from the followingexample in R2.

EXAMPLE 6.1. The saw-tooth function

f(x) = 2kg(2k(x 2k); x 2 2k; 21k; k = 1; 2; : : : ;

where g(x) = 1=2 j1=2 xj, does not belong to the closure of Q(R2) inW 1;1(S1).

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One may also consider a norm on Q(Rn) induced by the norm in Sobolev spacesW 1;p, 1 6 p i:Fix an i and let j increase, then choose H-converging subsequences from thesequences (Ci)j = Cij and (Di)j = Dij . We get Ai + Ci = A + Di and bothsequences kCik and kDik vanish as i!1, hence Ai BP! A as i!1.

Now, let us prove the statement of the theorem for D. Let fAig be a D-Cauchy sequence in Y . Again, it is H-convergent to a set A 2 Y . Denote byT the intersection of all T (Ai), i = 1; 2; : : :, and T (A); it is still a set of fullmeasure. Obviously, for any v 2 T ,

limi!1 kA(v) Ai(v)k = 0:

Moreover, this convergence is uniform for all v 2 T \ Sn1 because fAig is aD-Cauchy sequence. It follows from definitions that Ai

D! A as i!1. 2

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8. Derivatives

There are some definitions of differentials and derivatives for multivalued map-pings. The most frequently used ones are based on various form of contingentand tangent cones with respect to the graph of the mapping, see, for example[2, 14]. These definitions are very convenient for studying mappings definedby means of a system of inequalities and equalities. However it is difficult toapply them for other kinds of mappings arising in application. For example somemodels of economic dynamics are described by mappings of the following form:

a(x) = A(x) +B(x);

whereA(x) = f(x)A and B(x) = fy: F1(x) 6 y 6 F2(x)g: (8.1)

Here A is a convex compact subset of the cone Rn+ of all vectors with nonnegativecoordinates, f is a real-valued function defined on Rn+ and F1; F2: Rn+ ! Rn+with the property F1(x) 6 F2(x); it is supposed that the usual coordinate-wiseorder relation > is introduced in Rn+. See the book [24] devoted to the study ofmodels defined by means of mappings of the form (8.1). Different definitions ofderivative based on properties of the space Q often happen to be more convenientfor the study of such mappings.

We start with single-valued mappings F : X ! X, where X is a normedspace. The directional derivative F 0(x; u) of the mapping F at a point x in adirection u is defined by the formula

F 0(x; u) = limt!+0

1t(F (x+ tu) F (x)):

In order to define this derivative we need only the mapping t 7! F (x + tu)defined on the segment (0; t0) with small t0. So we restrict consideration tovector-valued functions x(), that is mappings defined on a open segment I ofthe real line and acting into X. The right-hand derivative x0(t) of a function xat a point t 2 I is defined by the formula

x0(t) = lim!+0

1

(x(t+ ) x(t)):The simplest example of a differentiable function is delivered by a functionx(t) = f(t)x, where f is a right-hand differentiable function and x is a fixedelement of X. Since any linear combination of differentiable functions is alsodifferentiable, it follows that a function of the form

x(t) =mXi=1

fi(t)xi

is differentiable. It is clear that x0(t) =Pmi=1 f

0i(t)xi, where f 0(t) is the right-

hand derivative of a function f .Denote by X 0 the conjugate space to X. The following assertion holds.

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PROPOSITION 8.1 (Weak mean value theorem). Let t1; t2 2 I; t1 < t2 andv 2 X 0

mv = inft2[t1;t2]

v(x0(t)); Mv = supt2[t1;t2]

v(x0(t)):

Thenmv(t2 t1) 6 v(x(t2) x(t1)) 6Mv(t2 t1):

Proof. It follows directly from the mean-value theorem for directionally dif-ferentiable functions [8] applied to the function t 7! v(x(t)). 2

COROLLARY 8.1. If x0(t) = 0 for t 2 I then x(t) = const.Now assume that X = Q with an arbitrary norm. Let p0(t) be the right-handderivative of a vector-valued function p: I ! P Q. Representing the elementp0(t) of the space Q as the difference of two sublinear functions p0+() and p0()we get that the function p() is differentiable at the point t if and only if thereexist functions p0+; p0 2 P such that

k(p(t+ ) + p0) (p(t) + p0+)k = o():Applying Minkowski duality we can define differentiability for functions froman open segment I into Y . Let Z be one of the metrics H; D or BP.

DEFINITION 8.1. We say that a pair (A+; A) 2 Y Y is the right-handderivative A0(t) of a function A: I ! Y at a point t with respect to the metricZ if

Z(A(t+ ) + A; A(t) + A+) = o():

Thus the derivative is the class of equivalent pairs since two pairs generate thesame difference of sublinear functions if and only if these pairs are equivalent.For the Hausdorff metric Z = H this definition was introduced by Yu. N. Tyurinin [29] and Banks and Jakobs in [4], see also [6, 21] for related definitions.

As follows from the hierarchy of the three metrics, the BP-differentiabil-ity implies the D-differentiability, and the latter implies the H-differentiability.Moreover, the same examples that show the difference between three metrics canbe used to prove that all three derivatives are indeed different.

Let us consider two examples.

EXAMPLE 8.1. Let A(t) =Pmi=1 fi(t)Ai, where real-valued functions fi are

right-hand differentiable on I and Ai 2 Y . It follows from above that the functionA is differentiable with respect to BP, and therefore with respect to H; D, and

A0(t) =mXi=1jf 0i(t)j(Ai+; Ai);

where Ai+ = Ai, Ai = f0g if f 0i(t) > 0 and Ai+ = f0g, Ai = Ai if f 0i(t) 6 0.

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EXAMPLE 8.2. Let F1; F2: I ! Rn+ be right-hand differentiable mappings, sothere exist elements y1; y2; o1(t); o2(t) such that, for > 0,

Fi(t+ ) = Fi(t) + yi + oi()

and oi()=! 0 as ! +0. Assume F1 6 F2 and define A(t) = fx: F1(t) 6x 6 F2(t)g. Here by x 6 y we mean xi 6 yi, i = 1; : : : ; n. Clearly pA(t)(v) =hv+; F2(t)ihv; F1(t)i, where v+ = max(v; 0); v = max(v; 0). It is easy tocheck that the mapping A is differentiable with respect to BP. If y1 > 0; y2 > 0then A0(t) = (A+; A), where

A = fx: 0 6 x 6 y1g; A+ = fx: 0 6 x 6 y2g:

The H-differentiability of this mapping was established in [4].

Examples 8.1 and 8.2 show that the mapping a defined by (8.1) is differentiablein YBP.

The following two lemmas are elementary consequences of Lemma 6.2 andusual properties of derivatives in normed spaces.

LEMMA 8.1. If A(t) has a right-hand Z-derivative (B+; B) at t = 0 and (t)is a real-valued function such that 0(0) exists, then the function C(t) = (t)A(t)has a right-hand Z-derivative at t = 0 and

C 0(t) =

(B+ + 0(0)A(0); B) if 0(0) > 0;

(B+; B + (0(0))A(0)) if 0(0) 6 0: (8.2)

LEMMA 8.2. If A(t) has a right-hand Z-derivative (B+; B) at t = 0 and ifC(t) has a right-hand Z-derivative (D+;D) at t = 0, then A(t) + C(t) has aright-hand Z-derivative (B+ +D+; B +D) at t = 0.

Note that a polyhedral-valued function F (t) can be H-differentiable and not D-differentiable at some points. Consider Example 3.1, where (t) = t2 makesS(t) H-differentiable at t = 0, but not D-differentiable.

Let YZ be the set Y with a metric Z, where Z is one of H; D; BP. For amapping A: I ! YZ we can define derivative by means of Demyanov difference.Namely, the DZ-derivative A0Z(t) of the mapping A at a point t is defined by theformula:

A0Z(t) = Z lim1

(A(t+ ) . A(t)):

PROPOSITION 8.2. LetA(t) be a YBP-differentiable mapping with the derivative(A+; A). Then the DH-derivative A0H(t) exists and A0H(t) = A+

. A.

SVAN329.tex; 10/11/1997; 12:47; v.7; p.15


Proof. We haveD(A(t+ ); A(t)) = (A+; A) + o()

= (A+ + +(); A + ())

and +(); ()D! 0. Applying Lemma 4.2 we get

A(t+ ) . A(t) = (A+ . A + ());where ()!H 0. 2

The Tyurin derivative induces a class Q of linear convex-valued functions asfunctions A(t): [0; a]! Y such that, for some B; B+ 2 Y

A(0) + tB+ = A(t) + tB; 0 6 t 6 a:

By definition, a function A() 2 Q satisfies alsoA(t) = (A(0) + tB+) . tB; 0 6 t 6 a: (8.3)

Denote the class of functions A(t) satisfying (8.3) by R.

PROPOSITION 8.3. The class R is strictly broader than Q.Proof. We shall show that there exist functions of the form A(t) = A . tB

which do not belong toQ. Indeed, this is, for example, the case for A = f(x; y) 2R2: jxj 6 1; jyj 6 1g and B = f(x; x) 2 R2: 0 6 x 6 g in R2, seeExample 5.2. 2

In conclusion of this section, let us notice that the three metrics H, D, and BPcan be used for the study of the conjugate derivative of set-valued mappings, see[21].

9. Convex Fuzzy Sets

A fuzzy set is an upper semi-continuous function U(x) from Rn onto [0; 1]. Its-level sets are defined as

[U ] = fx 2 Rn: U(x) > g for each 2 (0; 1]while its support is the set

[U ]0 =[

2(0;1][U ]:

DEFINITION 9.1. A fuzzy set is convex if all its level sets [U ], 0 6 6 1, areconvex.

SVAN329.tex; 10/11/1997; 12:47; v.7; p.16


In other words, a convex fuzzy set is an upper semi-continuous quasiconcavefunction U(x) from Rn to [0; 1], see Figure 2. It can be interpreted also as aY -valued function on [0; 1], antimonotone with respect to inclusion,

s

s

6

-

A

AAAAA

BBBBBBBBB

u

x

1

Figure 2. Convex fuzzy set.

The three metrics addressed in this paper generate corresponding structuresof a metric space on the set EK of fuzzy sets with a compact support K.

DEFINITION 9.2. Denote by dZ1 the following metric on the space E of convexfuzzy sets with a compact support in Rn:

dZ1(U; V ) = supt2[0;1]

Z([U ]t; [V ]t);

where Z is one of H, D, or BP.

The metric space dH1 was considered in [9]. The following assertion easily fol-lows from the definitions and Theorem 7.1.

PROPOSITION 9.1. The space E with the metric dZ1 is a complete metric space.Proof. Indeed, if Ui is a Z-Cauchy sequence of fuzzy sets then, for each

2 (0; 1] the sequence [Ui] Z-converges to a set V () 2 Y . Let us show thatthe sets V () are the level sets of a convex fuzzy set V . We have to provethat the mapping ! V () is antimonotone with respect to inclusion and thatits graph is closed in R Rn. The first part follows from the antimonotonicityof the mappings ! [Ui]. To prove the second part, notice that it is alreadyproved for Z = H, see [9], and both D- and BP-convergences are stronger thenthe Hausdorff one. 2

SVAN329.tex; 10/11/1997; 12:47; v.7; p.17


Denote the epigraph of a fuzzy set U by epiU = f(x; ): 6 U(x)g. For afuzzy set U , following Kloeden [13], we define the sendograph of U as

send (U) = f(x; ) 2 epiU : > 0g:First we consider fuzzy sets with convex sendographs. The notion of Moscoconvergence of functions which is also called epi-convergence is often used inconvex analysis. It is said that a sequence fn of quasiconcave functions convergesto a function f in the sense of Mosco if the epigraphs epi fn of fn converge to theepigraph of f in PainleveKuratowski sense, see [1, 15, 16, 27] for definitionsand discussion. Here epi f = f(x; ): 6 f(x)g.

Next, we shall study generalizations of Mosco convergence for fuzzy setswith convex sendographs, namely, we consider the convergence of sendographswith respect to the metrics H and D. For this purpose we need the followingdefinitions from convex analysis.

DEFINITION 9.3. Let f be a concave function defined on a convex compact setK. The function f defined on Rn by the formula

f(v) = minfhv; xi f(x): x 2 Kgis called YoungFenchel conjugate to f .

Let f be a finite function defined on the space Rn. A positively homogeneousextension f0 of the function f is the function defined on Rn (0;+1) by theformula f0(x; c) = cf(x=c). Another way to define f0 is to say that its epigraphin RnR is the convex cone hull of the epigraph of the function f(x; 1) f(x),x 2 Rn defined on Rn f1g.

Let U be a fuzzy set with convex sendograph SU . Let us calculate the supportfunction pSU . For points in Rn R, we define the inner product as follows:

h(v; c); (x; )i = hv; xi + c:We get

pSU(v; c) = maxx2[U ]0;066U(x)

hv; xi + c;

so for c 6 0 we have pSU(v; c) = p[U ]0(v). If c > 0 then

pSU(v; c) = maxx2[U ]0

(hv; xi + cu(x))

= c minx2[U ]0

Dvc; xE U(x)

= cU

vc

;

SVAN329.tex; 10/11/1997; 12:47; v.7; p.18


where [U ]0 is the support of U . Thus

pSU(v; c) = (U)0(v; c) (9.1)Let us now calculate the distance H(U; V ) = H(SU;SV ) between two fuzzy

sets U and V with convex sendographs. We have

H(U; V ) = max(v;c)2Sn

jpSU (v; c) pSB(v; c)j

= max

maxkvk61

jp[U ]0(v) p[V ]0(v)j;

sup0 0. Then pSU has a form (9.1). A simplecalculation shows that

rpSU(v; c) =rU

vc

; U

vc

+

1c2

DrU

vc

; vE: (9.2)

Thus for fuzzy sets U; V with convex sendographs we have

D(U; V ) = max(m1;m2);

where

m1 = supfrp[U ]0(v) rp[V ]0(v): 9c < 0such that (v; c) 2 TSU \ TSV g

and

m2 = supfrpSU (v; c) rpSV (v; c): (v; c) 2 TSU \ TSV ; c > 0g;where pSU and pSV are calculated by (9.2).

Clearly m1 = D([U ]0; [V ]0). We can also consider m2 as a distance betweenconjugate functions U and V calculated in a certain modification of the normW 1;1. Thus a sequence Uk converges to U in the sense of metric D if and onlyif D([Uk]0; [U ]0)! 0 and Uk converges to U in the above sense.

SVAN329.tex; 10/11/1997; 12:47; v.7; p.19


It is natural to expect that the distance dD1(U; V ) between two convex fuzzysets also can be expressed in the terms of conjugate functions. For this purpose,we need an appropriate notion of conjugacy for quasiconvex functions. In con-trast with convex analysis, there is a lot of different definitions of conjugacy inquasiconvex analysis (see, for example [18]). Since the class of all quasiconvexfunctions is extremely broad it is necessary to add one more dimension to thedomain of the conjugate function. In order to avoid the additional dimension,we are going to restrict consideration to special subclasses of the class of allquasiconvex functions. We shall apply in this paper the conjugation operation( ) defined in [25, 26]. This operation is defined on the set Qu0 of all low-er semicontinuous nonegative quasiconvex functions U : Rn ! R+1 such thatU(0) = 0. By definition,

U(v) = suphv;xi>1

1U(x)

:

It is assumed that supremum on the empty set is zero. The equality U = Uholds for U 2 Qu0. Let [U ]c = fx: U(x) 6 cg and (U)c = fx: U(x) < cg. Itcan be shown (see [25]) that, for c > 0,

[U] 1c

= ((U)c); (U) 1

c=[c0>c

([U ]c0)

where ( ) denotes the polar set.A vector v 2 Rn is called a subgradient of a function U 2 Qu0, at a point x

such that U(x) > 0 if U(v)U(x) = hv; xi = 1. We need the following assertion,see [26].PROPOSITION 9.3. Let U be a continuous function, U 2 Qu0, x 2 Rn andcl (U)t = [U ]t with t = U(x) > 0. Then a vector v is a subgradient of U at thepoint x if and only if

hv; xi = maxx02[U ]t

hv; x0i = 1:

A vector v is called the normalized generalized gradient of a function U 2 Qu0at a point x with U(x) > 0 if v is the only subgradient of the function U at thepoint x. We shall denote the normalized generalized gradient by brU(x). Thefollowing proposition explains this term.

PROPOSITION 9.4. If a function U has a gradient rU(x) at a point x andhrU(x); xi 6= 0 then there exists the normalized generalized gradient brU(x)and

brU(x) = rU(x)p[U ]t(rU(x))

with t = U(x).

SVAN329.tex; 10/11/1997; 12:47; v.7; p.20


Proof. Let v = rU(x). For y 2 [U ]t, we have0 > U(y) t = U(y) U(x) = hv; y xi+ o(ky xk);

hence

hv; xi = maxfhv; yi: y 2 [U ]tg = p[U ]t(v):It is easy to check that the equality hu; xi = p[U ]t(u) implies u = v with > 0so there exists brU(x) and brU(x) = v for some > 0. By definition, we have1 = h brU(x); xi = hv; xi = p[U ]t(v). 2In order to apply the conjugation operation () to convex fuzzy sets we shallconsider a function U 0 = 1 U instead of a convex fuzzy set (i.e concavefunction) U . Clearly, U 0 is a lower semicontinuous quasiconvex and nonnegativefunction. So, if 0 2 [U ]0 and U(0) = 1 then U 0 2 Qu0. Further we shallonly consider fuzzy sets U with the property U(0) = 1, so that we can apply theconjugation operation (). Clearly, [U 0]t = [U ]1t for any fuzzy set U . Therefore,dZ1(U; V ) = dZ1(U 0; V 0) for Z = H;D;BP.

Let Qu1 = fU 2 Qu0: U(0) = 0g. We shall study the restriction of themetric dD1 to the set Qu1. The following assertion holds.

PROPOSITION 9.5. Let U 2 Qu1 be a continuous function with the propertycl (U)t = [U ]t for all 0 < t 6 1. Then a vector x 6= 0 is a gradient of the supportfunction p[U ]t for t 6= 0 at a point v 2 T[U ]t if and only if x = brU(v) where = (p[U ]t(v))

1.

Proof. Let v 2 T[U ]t , that is there exists a unique vector x 2 [U ]t such thathv; xi = max

x02[U ]thv; x0i: (9.3)

If x 6= 0 then hv; xi > 0. Indeed, since 0 2 [U ]t, we have > 0. If = 0 then the linear function x 7! hv; xi achieves its maximum over the set[U ]t at least at two points x1 = x and x2 = 0. Since v 2 T[U ]t , we cometo a contradiction. Thus > 0. Let = 1=. It follows from Proposition 9.3that v is a subgradient of the function U at the point x, so U(v)U(x) = 1.This equality shows that x is a subgradient of the function U at the point v.Assume now that y is a subgradient of U at the point v. Then the equalityU(v)U(y) = 1 shows that U(y) = U(x) = t, therefore y 2 [U ]t. We havealso hv; yi = 1 hence

hv; yi = 1

= hv; xi = maxx02[U ]t

hv; x0i:

The inclusion v 2 T[U ]t shows that y = x. Thus x = brU(v). It also followsfrom (9.3) that 1= = p[U ]t(v).

SVAN329.tex; 10/11/1997; 12:47; v.7; p.21


Let now x = brU(v) with 1= = p[U ]t(v). As follows from Proposition9.3, x is the only vector such that U(x) = t and

U(v)U(x) =1

= hv; xi = p[U ]t:

Thus v 2 T[U ]t and x = rp[U ]t(v). 2

Next, let us show that it is possible, under some assumptions, to apply normalizedgeneralized gradients in order to express the distance dD1(U 0; V 0) between convexfuzzy sets U 0; V 0 in the uniform Demyanov metric dD1. We shall consider onlycontinuous fuzzy sets U 0; V 0 with the property U 0(0) = V 0(0) = 1. Assume that

cl (U 0)t = [U 0]t; cl (V 0)t = [V 0]t:

Let U = 1 U 0; V = 1 V 0. By definition,dD1(U

0; V 0) = dD1(U; V ) = sup06t61

D([U ]t; [V ]t)

= max(m1;m2);

where

m1 = D([U ]0; [V ]0)

and

m2 = sup0


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comparative properties of three metrics in the space of compact convex sets

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