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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bigcap_{i\in I}R_{i}$\end{document} to agree with the supremum of the infima of f over the Ri’s.


Introduction
Let S be a set, (R i ) i∈I be a family of subsets of S, C = i∈I R i its union, R = i∈I R i its intersection and E be a complete ordered space. Given a map f : C −→ E, we are interested in finding conditions which ensure the equality (1.1) space, C is a convex subset of V , E = R, f is a quasiconvex function and the family (R i ) i∈I satisfies certain conditions, basically R i ∩ R j = R (i = j ). These results were applied to the case when S = X is a normed space, C ⊂ X is a convex subset, x ∈ 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 d(x, R) = sup i∈I d(x, R i ).
(1.2) Equality (1.2) has been studied by many authors in different situations (Castillo and Papini 2007;Chacón et al. 2004;Hoffmann 1992;Martínez-Legaz and Martinón 2007;Martínez-Legaz et al. 2002;Martinón 2004;Rubinov 2003;. In this paper we obtain analogous results to those of Martínez-Legaz and Martinón (2011) for the case when f is an order bounded quasiconvex vector function taking values on an order complete Riesz space E. We observe some differences between the scalar and vector cases. Indeed, under certain conditions, the equality holds for every i ∈ I except at most one if f is scalar; however, under similar requirements, it is possible to have for every i ∈ I if f is a vector function, although (1.1) is still true. If E ⊂ R D is a complete function space, for example if the positive cone of E is a Yudin cone (see definitions in Sect. 5), it is possible to obtain better results: the equality (1.3) holds for any i ∈ I \ I D , where the cardinality of I D is less than or equal to the cardinality of D.
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 on the sets R i , the distance from a point x to the intersection i∈I R i coincides with the supremum of the distances from x to each R i . In Kusraev and Kutateladze (2006) 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 satisfies the following properties (Aliprantis and Tourky 2007, Definition 1.2): (1) K + K ⊂ K; (2) αK ⊂ K, for all scalars α ≥ 0; (3) K ∩ (−K) = {0}. In this case, K defines an order x ≤ y ⇐⇒ y − x ∈ 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 ∈ E has a maximum and a minimum we say that E is a Riesz space. Throughout this paper E denotes an order complete Riesz space; that is, E is a Riesz space such that any order bounded subset A of E has a supremum sup A and an infimum inf A. Recall that the infinite distributive laws are valid (Aliprantis and Tourky 2007, Exercise 1.3.11;Luxemburg and Zaanen 1971, Theorem 2.12.2): given a ∈ E and a family (a i ) i∈I of vectors of E, we have inf a, sup 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 ∈ C and 0 ≤ t ≤ 1, we obtain equivalently, for every e ∈ E, the set {x ∈ C : f (x) ≤ e} is convex (Luc 2005, Proposition 5.9).
Several authors (see, for example, Chen et al. 2005) introduce the following definition: given linear spaces V and W , a convex cone K = {0} in W and a nonempty subset C of V , the map f : C −→ W is called quasiconvex if, for any b ∈ W , the set {a ∈ C : f (a) ∈ b − K} 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 definitions 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 ∈ E, the set M e = {x ∈ C : f (x) < e} is convex. B. For every e ∈ E, the set N e = {x ∈ C : f (x) ≤ e} is convex.
If E = R, then it is well known that A ⇐⇒ B. In the general complete Riesz space case we have A =⇒ B. In fact, for every e ∈ E, the equality N e = g>e M g holds. Indeed: for every g > e, hence N e ⊂ g>e M g ; on the other hand, if x ∈ g>e M g , then, for every g > e, we have f (x) < g, so f (x) is a lower bound of {g : g > e}, hence f (x) ≤ inf{g : g > e} = e. As the sets M g are convex, we find that N e is convex.
The implication B =⇒ A is not valid, as shown in the next example.
As f 1 and f 2 are quasiconvex, we see that f is quasiconvex. Moreover, the set

Separation
Let V be a vector space. Denote by [x, y] the segment with endpoints x and y, that is, [x, y] For nonempty subsets R 1 , R 2 , R of V , we say that R separates the sets R 1 and R 2 if [r 1 , r 2 ] ∩ R = ∅, for every r 1 ∈ R 1 and r 2 ∈ R 2 (Pallaschke and Urbański 2002, p. 489).
We generalize the above notion in the following way: for a family (R i ) i∈I of nonempty subsets of V containing at least two members, we say that a subset R of V separates the family (R i ) i∈I if R separates the sets R i and R j for every i, j ∈ I , i = j . Notice that in this situation i∈I R i ⊂ R.
In Pallaschke and Urbański (2002, Proposition 2.3) it is proved that if A, B are nonempty convex algebraic closed subsets of a vector space V , the set A∩B separates the sets A and B if and only if the union A ∪ B is a convex set. Observing that the proof of the "only if" part of that statement does not use the algebraic closedness assumption, one easily gets the following result: given a family (R i ) i∈I of nonempty convex subsets of a vector space V such that i∈I R i separates (R i ) i∈I , we see that R j ∪ R k is convex for every j, k ∈ I .
Similarly to Pallaschke and Urbański (2002, Proposition 2.3), the following result characterizes convexity in terms of separation. We leave the simple proof to the reader.
Proposition 3.1 Let C be a nonempty subset of a vector space V . The following assertions are equivalent: (1) C is convex.
(2) There exists a family (R i ) i∈I of nonempty convex subsets of V containing at least two members such that i∈I R i = C and i∈I R i separates (R i ) i∈I . (3) There exists a family (R i ) i∈I of nonempty convex subsets of V containing at least two members such that i∈I R i = C and R j ∪ R k is convex for every j, k ∈ I .

The infimum over an intersection
In this section we give the main results of this paper: under certain conditions we see that equality (1.1) holds.
Proposition 4.1 Let V be a vector space, C ⊂ V be a convex set, (R i ) i∈I be a family of nonempty subsets of C containing at least two members, R := i∈I R i , E be an order complete Riesz space, and f : Hence equality (1.1) holds.
Fix r i ∈ R i , apply the infinite distributive laws (2.1) and obtain for all i ∈ I , from which we obtain the converse inequality; so (4.1) is proved. Equality (1.1) follows from (4.1) 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 finite, under the assumptions of the preceding proposition one may have inf x∈R f ( Example 4.2 Let L denote the vector subspace of R [0,1] consisting of all continuous functions of bounded total variation, and consider the cone then L is order complete (Aliprantis and Tourky 2007, Problem 1.3.20). We write for any x ∈ [0, 1]. Notice that the order associated to K is defined by We take the functions a, b, c ∈ L defined in the following way: The function f is order bounded and quasiconvex, because from sup{a, 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 ) i∈N be a sequence of nonempty subsets of C consisting of linearly closed sets in C such that i∈N R i = C and R i ∩ R j = R, for i, j ∈ N with i = j , E be an order complete Riesz space, and f : C −→ E be an order bounded quasiconvex function. Then R = ∅ and, for i, j ∈ N with i = j, Proposition 4.4 Let V be a vector space, C ⊂ V be a convex set, E be an order complete Riesz space, f : C −→ E be an order bounded quasiconvex function, R ⊂ C, R 1 , R 2 , . . . , R n ⊂ C (n ≥ 2) be nonempty and linearly closed in C satisfying n i=1 R i = C and such that, for a certain h = 1, . . . , n and for every j = 1, . . . , n (4.3) Then R = ∅ and due to (4.3). From this we obtain the result.

Function spaces
If E is a coordinate space we can work with the components of the function f . More precisely, in Aliprantis and Border (2006, Consider now a convex subset C of a vector space V . Given a function f : C −→ E, we consider its components f d (d ∈ D): It is easy to prove that f is quasiconvex ⇐⇒ ∀d ∈ D, f d is quasiconvex and f is order bounded ⇐⇒ ∀d ∈ 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 differences between the scalar and vector cases, we first recall the following scalar result: Proposition 5.1 (Martínez-Legaz and Martinón 2011, Proposition 3.1) Let V be a vector space, C ⊂ V be a convex set, (R i ) i∈I be a family of nonempty subsets of C containing at least two members, R := i∈I R i and f : for every d ∈ D and i ∈ I \ I D . This shows (5.2).
In the vector case it is not possible to guarantee equality (5.1) for every i ∈ I except only one i, as we show in the next examples.
, which is quasiconvex and order bounded. Let It is clear that R separates the family (R −1 , R 0 , R 1 ). Then Consequently, Therefore, inf x∈R f (x) coincides with only one of the values inf x∈R k f (x). 1, 2, 3, . . .).

Consequently, none of the values inf x∈R k f (x) coincides with inf x∈R f (x).
A cone K of a vector space E is called a Yudin cone if K is generated by a Hamel Basis (e d ) d∈D of E; that is, K is the smallest cone that includes all the e d (d ∈ D) (Aliprantis and Tourky 2007, Definition 3.15). If K is a Yudin cone of the vector space E, then E ordered by K is an order complete Riesz space (Aliprantis and Tourky 2007, Theorem 3.17 that is, E = c 00 (D) (Aliprantis and Tourky 2007, 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 ) i∈N be a sequence of nonempty subsets of C consisting of linearly closed sets in C such that for some set D = ∅, and f : C −→ E be an order bounded quasiconvex function. Then R = ∅ 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 (5.2) for every k ∈ N \ N D .

A topological result
In Martínez-Legaz and Martinón (2007, Theorem 5) it is proved that if X is a metric space with distance d, C ⊂ X is a closed subset and x ∈ X, then the following assertions are equivalent: (1) C is x-boundedly connected; that is, for every ε > 0, the set {y ∈ C : d(y, x) < ε} is connected.
(2) If R and S are closed subsets of X with R ∪ S = C, then the equality d(x, R ∩ S) = max{d(x, R), d(x, S)} holds.
The next result is an analog 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 {x ∈ X : e ≤ f (x)} is closed for any e ∈ E. Proposition 6.1 Let X be a topological space, E be an order complete Riesz space, and f : X −→ E be an upper level closed and order bounded function. The following assertions are equivalent: (ii) If R and S are nonempty closed subsets of X and R ∪ S = X then at least one of the equalities But this is impossible, since every y ∈ R ∩ S satisfies e = inf x∈R∩S f (x) ≤ f (y) and hence does not belong to U . (ii) =⇒ (i). Suppose that U := {x ∈ X : e ≤ f (x)} is not connected for some e ∈ E. Then there exist two closed sets R, S ⊂ X such that R ∩ U = ∅, S ∩ U = ∅, ( R ∪ S) ∩ U = U and R ∩ S ∩ U = ∅. Define R := R ∪ (X \ U) and S := S ∪ (X \ U). These sets are closed; moreover, On the other hand, since R ∩ S ∩ U = ∅, for every x ∈ R ∩ S one has x / ∈ U, that is, e ≤ f (x); therefore e ≤ inf x∈R∩S f (x), which, together with (6.2), contradicts (ii).

Application to the distance function on a Kantorovich space
In this last section we apply the results of the above sections to the distance function defined on a vector (or cone) normed space or Kantorovich space (Kusraev 1993). Recently research in Kantorovich spaces has had a great development: see, for example (Kusraev 1985;Sonmez andCakalli 2010 andZabrejko 1997). In Kusraev and Kutateladze (2006) the interaction between certain optimization problems and such spaces is shown.
Let us begin with the definition: let X be a vector space and E be an order complete Riesz space. A map · : X −→ E is said to be a vector norm (or cone norm) on X if, for x, y ∈ X and α ∈ R, (1) x ≥ 0, (2) x = 0 ⇐⇒ x = 0, (3) αx = |α| x , (4) x + y ≤ x + y .
In this situation we say that X endowed with · is a vector normed space (or cone normed space, or Kantorovich space). From a vector norm · we can derive a vector distance (or cone distance) d(x, y) = x − y which satisfies the usual properties of a distance.
Given a vector normed space X with · taking values in a order complete Riesz space, if x ∈ X and ∅ = C ⊂ X, then one defines the vector d(x, C) = inf y∈C d(x, y).
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 · taking values on an order complete Riesz space. Let C ⊂ X be a convex set, (R i ) i∈I be a family of nonempty subsets of C containing at least two members and R := i∈I R i . If R separates (R i ) i∈I , then, for i, j ∈ I with i = j, for every x ∈ X. Hence equality (1.2) holds.