On the infimum of a quasiconvex vector function over an intersection

We give sufficient conditions for the infimum of a quasiconvex vector function f over an intersection $\bigcap_{i\in I}R_{i}$ to agree with the supremum of the infima of f over the Ri’s.


Introduction
Let S be a set, (R i ) i2I be a family of subsets of S, C = S i2I R i its union, R = T i2I R i its intersection and E be a complete ordered space. Given a map f : C ! E, we are interested in …nding conditions which ensure the following equality: (1.1) Of course, the inequality is always satis…ed.
In [13] we have considered several settings. Our main results about equality (1.1) were obtained in the case when S = V is a linear space, C is a convex subset of V , E = R, f is a quasiconvex function and the family (R i ) i2I satis…es certain conditions, basically R i \ R j = R (i 6 = j). These results were applied to the case when S = X is a normed space, C X is a convex subset, x 2 X is a given point, and f is the distance function d(x; ) : C ! R, which is convex, hence quasiconvex, and we obtained the equality Equality (1.2) has been studied by many authors in di¤erent situations [3,4,6,12,14,15,17,18,19].
In this paper we obtain analogous results to those of [13] for the case when f is an order bounded quasiconvex vector function taking values on an order complete Riesz space E. We observe some di¤erences between the scalar and vector cases. Indeed, under certain conditions, the equality These general results are applied to the vector distance (or cone distance) derived from a vector norm (or cone norm) and to obtain the validity of the equality (1.2) for vector distances. That is, in the vector case we prove that, under some requirements about the sets R i , the distance from a point x to the intersection T i2I R i coincides with the supremum of the distances from x to each R i . In [9] vector normed spaces, also called Kantorovich spaces, and their relation with certain optimization problems are considered.

Quasiconvex vector functions
A nonempty subset K of a (real) vector space E is said to be a pointed convex cone if it satis…es the following properties [2, De…nition 1.2]: (1) K + K K; (2) K K, for all scalars 0; (3) K \ ( K) = f0g. In this case, K de…nes an order x y () y x 2 K on E which is compatible with its vector structure and we say that E is an ordered vector space. If every pair x; y 2 E has a maximum and a minimum we say that E is a Riesz space. inffa; a i g ; supfa; inf supfa; a i g : The next well-known notion is basic in this paper: Let V be a linear space, C V a convex set and E be a Riesz space. The function f : C ! E is called quasiconvex if, for every x; y 2 C and 0 t 1, we obtain equivalently, for every e 2 E, the set fx 2 C : f (x) eg is convex ([10, Prop. 5.9]).
Several authors (see, for example, [5]) introduce the following de…nition: given linear spaces V and W , a convex cone K 6 = f0g in W and a nonempty subset C of V , the map f : C ! W is called quasiconvex if, for any b 2 W , the set fa 2 C : f (a) 2 b Kg is a convex set. If K is a pointed convex cone, hence W is an ordered vector space, and if moreover W is a Riesz space and C is convex, then both de…nitions coincide.
Remark 2.1. Assume that V is a linear space, C V a convex set, E a Riesz space and f : C ! E a map. Consider the following assertions: A. For every e 2 E, the set M e = fx 2 C : f (x) < eg is convex.
B. For every e 2 E, the set N e = fx 2 C : f (x) eg is convex.
hence N e T g>e M g ; on the other hand, if x 2 T g>e M g , then, for every g > e, we have f (x) < g, so f (x) is a lower bound of fg : g > eg, hence f (x) inffg : g > eg = e. As the sets M g are convex, we obtain that N e is convex.
The implication B =) A is not valid, as shown in the next example.
x 1 As f 1 and f 2 are quasiconvex, we have that f is quasiconvex. Moreover, the set is not convex.

Separation
Let V be a vector space. Denote by [x; y] the segment with endpoints x and y, that is, [x; y] := ftx + (1 t)y : 0 t 1g.
We generalize the above notion in the following way: for a family (R i ) i2I of nonempty subsets of V containing at least two members, we say that a subset R of V separates the family (R i ) i2I if R separates the sets R i and R j for every i; j 2 I, i 6 = j. Notice that in this situation Similarly to [16,Proposition 2.3], the following result characterizes convexity in terms of separation. We leave the simple proof to the reader. (1) C is convex.
(2) There exists a family (R i ) i2I of nonempty convex subsets of V containing at least two

The infimum over an intersection
In this section we give the main results of this paper: under certain conditions we obtain that equality (1.1) holds.
be a family of nonempty subsets of C containing at least two members; R := T i2I R i , E be an order complete Riesz space, and f : C ! E be an order bounded quasiconvex function.
Proof. Let i; j 2 I with i 6 = j, and let r i 2 R i and r j 2 R j . There exists z 2 [r i ; r j ] \ R. As f is for all i 2 I, from which we obtain the converse inequality; so (4.5) is proved. Equality (1.1) follows from (4.5) and the assumption that the index set I has at least two elements.
The following example shows that, even in the case when the index set I is …nite, under the assumptions of the preceding proposition one may have inf x2R f (x) 6 = inf x2Ri f (x) for every i 2 I: Notice that the order associated to K is de…ned by f g () f g and g f is increasing: We take the functions a; b; c 2 L de…ned in the following way: x 1: The function f is order bounded and quasiconvex, because from supfa; cg (x) = 2x for all x 2 [0; 1] it follows that b supfa; cg. Consider the sets R 1 = [0; 1=2] and R 2 = [1=2; 1], so R = R 1 \ R 2 = f1=2g and R separates R 1 and R 2 . We obtain One has The next proposition deals with sequences of linearly closed sets. We recall that a subset S of a set C in a vector space V is said to be linearly closed in C if the intersection of S with every straight line L is closed in C \ L; this latter set being endowed with its natural topology as a subset of the straight line L.
Proposition 4.3. Let V be a vector space, C V be a convex set, R C; (R i ) i2N be a sequence of nonempty subsets of C consisting of linearly closed sets in C such that S i2N R i = C and R i \ R j = R, for i; j 2 N with i 6 = j, E be an order complete Riesz space, and f : C ! E be an order bounded quasiconvex function. Then R 6 = ; and, for i; j 2 N with i 6 = j; If < 1, for every p = 1; 2; 3::: there exists y p 2 S such that y p := (1 p )r + p s, being < p + 1=p. As S is the …nite union of the sets R k \ [r; s], there is a subsequence (y pm ) m 1 of (y p ) p 1 contained in some it is clear that t is the limit of a sequence of points of R k \ L and, consequently, t 2 R k ; hence Assume that E is a complete function space with E R D . Let (g j ) j2J be a family of functions Consider now a convex subset C of a vector space V . Given a function f : C ! E, we consider It is easy to prove that f is quasiconvex () 8d 2 D; f d is quasiconvex and f is order bounded () 8d 2 D; f d is bounded .
We will next improve Proposition 4.1 in the case E is a complete function space. In order to stress the di¤erences between the scalar and vector cases, we …rst recall the following scalar result: It is clear that R separates the family (R 1 ; R 0 ; R 1 ). Then Consequently, Therefore, inf x2R f (x) coincides with only one of the values inf x2R k f (x). Then [r i ; r j ] \ R i \ R j 6 = ;, for every r i 2 R i and r j 2 R j (i; j = 1; 2; 3::: with i 6 = j), and C = S 1 k=0 R k ; moreover R = T 1 k=0 R k and R i \ R j = R for i; j = 1; 2; 3::: with i 6 = j). Let The being 0 = f k (x). Then, for every n = 1; 2; 3::: and every k 6 = n, Hence, for every k = 1; 2; 3:::, Consequently, none of the values inf x2R k f (x) coincides with inf x2R f (x). that is, E = c 00 (D) [2, p.129]. Hence we can consider a vector space E endowed with a Yudin cone K as a complete function space and then we can apply the above result.
The following proposition can be proved in analogous way to Proposition 5.2 Proposition 5.5. Let V be a vector space, C V be a convex set, R C; (R i ) i2N be a sequence of nonempty subsets of C consisting of linearly closed sets in C such that S i2N R i = C and R i \ R j = R, for i; j 2 N with i 6 = j, E be a complete function space with E R D , for some set D 6 = ;, and f : C ! E be an order bounded quasiconvex function. Then R 6 = ; and the equality (1.1) holds. Moreover, for some set N D N with cardinality less than or equal to the cardinality of D one has (5.9) for every k 2 N n N D .

A topological result
In [12,Theorem 5] it is proved that if X is a metric space with distance d, C X is a closed subset and x 2 X, then the following assertions are equivalent: (1) C is x-boundedly connected; that is, for every " > 0, the set fy 2 C : d(y; x) < "g is connected.
(2) If R and S are closed subsets of X with R [ S = C, then the equality d(x; R \ S) = maxfd(x; R); d(x; S)g holds.
The next result is an analogue to the above one. If X is a topological space and E is an ordered linear space, the map f : X ! E will be called upper level closed if the upper level set fx 2 X : e f (x)g is closed for any e 2 E.  On the other hand, since R \ S \ U = ;; for every x 2 R \ S one has x = 2 U; that is, e f (x) ; therefore e inf x2R\S f (x) ; which, together with (6.14), contradicts (ii). A QUASICONVEX VECTOR FUNCTION OVER AN INTERSECTION   11 7. Application to the distance function on a Kantorovich space

ON THE INFIM UM OF
In this last section we apply the results of the above sections to the distance function de…ned on a vector (or cone) normed space or Kantorovich space [8]. Recently research in Kantorovich spaces has had a great development: see, for example, [7], [20] and [22]. In [9] the interaction between certain optimization problems and such spaces is shown.
Let us begin with the de…nition: let X be a vector space and E be an order complete Riesz space. A map k k : X ! E is said a vector norm (or cone norm) on X if, for x; y 2 X and 2 R, (1) kxk 0; (2) kxk = 0 () x = 0; (3) k xk = j jkxk; (4) kx + yk kxk + kyk: In this situation we say that X endowed with k k is a vector normed space (or cone normed space, or Kantorovich space). From a vector norm k k we can derive a vector distance (or cone distance) d(x; y) = kx yk which satis…es the usual properties of a distance. If C is convex, then d(x; ) is convex, hence quasiconvex, for any x.
The propositions of the previous sections can be applied in this context. For example, from Proposition 4.1 we obtain the next result: Proposition 7.1. Let X be a vector normed space with norm k k taking values on an order complete Riesz space. Let C X be a convex set, (R i ) i2I be a family of nonempty subsets of C containing at least two members and R := T i2I R i . If R separates (R i ) i2I , then, for i; j 2 I with i 6 = j; d(x; R) = supfd(x; R i ); d(x; R j ) ; for every x 2 X. Hence equality (1.2) holds.
Acknowledgments The authors are thankful to the referees for their helpful comments.