Between Quasi-convex and Convex Set-valued Mappings
Joël Benoist, Nicolae Popovici
To cite this version:
Joël Benoist, Nicolae Popovici. Between Quasi-convex and Convex Set-valued Mappings. Applied Mathematics Letters, Elsevier, 2004, 17, pp.245-247. .
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Between Quasi-Convex and Convex
Set-Valued Mappings
J. BENOIST
LACO, UPRESSA 6090, Department of Mathematics
University of Limoges, 87060 Limoges, France
benoist~unilim.fr
N. POPOVICI
Faculty of Mathematics and Computer Science
Babe§-Bolyai University of Cluj, 3400 Cluj-Napoca, Romania
popovici~math.ubbcluj.ro
Abstract-The aim of this paper is to give sufficient conditions for a quasi-convex set-valued
mapping to be convex. In particular, we recover several known characterizations of convex real-valued
functions, given in terms of quasiconvexity and Jensen-type convexity by Nikodem [1], Behringer [2],
and Yang, Teo and Yang [3].
Keywords-Generalized convexity, Set-valued mappings.
1. INTRODUCTION
Throughout the paper, we denote by X a linear space and by Y a topological linear space,
partially ordered by a closed convex cone K having a nonempty interior in Y. Let F: D ~ 2Y
be a set-valued mapping, defined on a nonempty convex subset D of X.
Recall that F is said to be K -convex if the inclusion
tF(x)
+ (1- t)F(x')
C F(tx + (1- t)x')
+K
(1)
holds for all x, x' E D and for every t E [0, 1].
By analogy to vector-valued functions, we say that F is K -quasi convex if for each y E Y the
level set LF(Y) := {x E D: yE F(x) + K} is convex. Since K is a convex cone, it can be easily
seen that F is K-quasiconvex whenever it is K-convex.
In order to get sufficient conditions for a K-quasi-convex mapping to be K-convex, we shall
consider the following concept of generalized convexity: F will be called weakly K -convex with
respect to a nonempty setT c]O, 1[ if for all x,x' E D there exists some t ET for which (1)
holds.
Note that this concept extends several notions of generalized convexity, which were intensively
studied in the literature in the particular case of real-valued functions. Indeed, if T C ]0, 1[ is a
1
singleton, we recover the Jensen-t ype convexity (see, e.g., [4] and referenc
es therein) , which is
nowadays also known as nearly convexity (see, e.g., [5]). On the other hand,
for T = ]0, 1[ we
recover the notion of weakly convexity, introduc ed by Aleman in [6]. As an
intermed iate case, if
T = [b', 1 - b'] with b' E ]0, 1/2[, we recover the notion of uniform convexlikeness,
which has been
introduc ed by Hartwig in [7].
Our aim here is to study the set-valued mappings, but our main result also
focuses on vectorvalued functions. Actually, if f : D --+ Y is a function defined on a nonemp
ty convex subset D
of X, then f will be called K-conve x (respectively, K-quasiconvex, or
weakly K-conve x with
respect to a nonemp ty set T c ]0, 1 [) if and only if the set-valued mapping
F : D --+ 2Y, defined
by F(x) = {f(x)} for all x E D, is K-conve x (respectively, K-quasiconvex,
or weakly K-conve x
with respect to T).
2. MAIN RESU LT
THEOREM 2.1. Let F : D
--+ 2Y be a set-valu ed mappin g defined on a nonemp
ty convex subset D
of X. If F has K-closed values (i.e., F(x) +K is a closed set for every x E
D), then the following
assertions are equivalent:
(i) FisK-c onvex;
(ii) F is both K-quasi convex and weakly K-conve x with respect to a nonemp
ty compact set
Tc]0,1 [.
PROOF. Obviously (i) implies (ii), the conclusion being true for any nonemp
ty setT c]O, 1[.
Conversely, suppose that (ii) holds and let T be a nonemp ty compac t subset
of]O, 1[ for which F
is weakly K-convex. Let us denote, for all x, x' E D,
Tx,x' := {t E [0, 1] : (1) holds}.
In order to prove (i), we just have to show that Tx,x' = [0, 1] for all x,x'
E D. To this end,
consider two arbitrar y points x 0 ,x1 E D and let us first prove that Tx
,x
0
1 is dense in [0, 1].
Suppose on the contrary that this is not the case. Then there exist some a,
bE [0, 1], a< b, such
that
[a, bJ n Txo,Xl
=
0.
(2)
f3 := inf[b, 1] n Txo,Xl.
(3)
Since {0, 1} C Txo ,x 1 , we can define the real numbers
and
Obviously, a:::; a < b:::; f3 and, by (2) and (3), we have
]a, (3[ n Txo,Xl = 0.
(4)
Let us denote, for all t E [0, 1],
Xt
:= txo
+ (1 -
t)x1
yt := tF(xo)
and
+ (1- t)F(x1).
By (3) and taking into account that T is compact , we can find some numbers
u E [0, a] n Tx 0 ,x 1
and v E [(3, 1] n Tx 0 ,x 1 such that tu+ (1 - t)v E ]a, (3[ for all t E T. On the
other hand, since F is
weakly K-conve x with respect toT, we can choose a number T E Txu,xv n
T. Hence,
/:=TU + (1- T)v E ]a,(J[.
Since 1 -1
= T(1- u) + (1- T)(1- v),
by definition of Yl' we can deduce that
Yl' =[Tu+ (1- T)v]F(x 0 ) + [T(1- u) + (1- T)(1- v)]F(x1)
c T[uF(x 0 ) + (1- u)F(x 1)] + (1- T)[vF(xo) + (1- v)F(x1)]
=
TYu
(5)
+ (1- T)Yv.
2
(6)
On the other hand, since u, v E Txa,xp we have Yu C F(xu) +K and Yv C F(xv).+K, and hence,
TYu + (1- T)Yv C TF(xu) + (1- T)F(xv) + K.
(7)
Recalling that T E Txu,xv, i.e., TF(xu) + (1- T)F(xv) C F(TXu + (1- T)xv) + K, by (6) and (7)
we infer that Y'Y C TYu + (1- T)Yv C F(Txu + (1- T)xv) + K = F(x'Y) + K, which means that
1 E Tx 0 ,x 1 • By (5) it follows that ]o:,,B[nTx0 ,x 1 -=f. 0, contradicting (4).
So, we have proved that Tx 0 ,x 1 is dense in [0, 1]. Now, let us show that Tx 0 ,x 1 = [0, 1]. Obviously,
{0, 1} C Txo,x 1 C [0, 1]. Consider an arbitrary t E]O, 1[. We just need to prove that t E Tx 0 ,xp
i.e., yt c F(xt) + K. Let y E yt. By definition of yt we have y = tzo + (1 - t)zl for some
z0 E F(xo) and z1 E F(x 1 ). Consider a point e E intK. By density of Tx 0 ,x 1 in [0, 1], we infer
the existence of two sequences: (t~)nEN in Tx 0 ,x 1 n [0, t] and (t;i;)nEN in Tx 0 ,x 1 n [t, 1], such that
1
{y~, y;i;} c y + -e- int K,
for all n;:::: 1,
n
where y~ = t~zo + (1- t~)z 1 and y;i; = t;t,"zo + (1- t;t,")z1. Then, we haye
1
y + -e E t:;;F(x 0 ) + (1- t~)F(x 1 ) +intK C F (xc) + K +intK C F (xt-) + K,
n
n
n
y+
~e E t;t F(x 0 ) +
(1- t;t)F(x 1 ) + int KC F (xt;\:) + K + int K C F (xt;\:) +I},
implying that {xt;;-, xt;\:} C Lp(y + (1/n)e), for all n;:::: 1. Recalling that F is K-quasiconvex and
taking into account that Xt E [xt;;-, Xt;t] for each n E N, we can deduce that
Xt
E
Lp
(y +~e) ,
i.e.,
y+
~e E F(xt) + K,
Finally, by letting n ___, oo, we infer that yE F(xt) + K = F(xt) + K.
for all n;:::: 1.
I
CoROLLARY 2.2. Let f : D ___, Y be a function defined on a nonempty convex subset D of X.
Tben fisK-convex if and only if it is botb K-quasiconvex and weakly K-convex witb respect
to a nonempty compact setT C ]0, 1[.
PROOF. It follows by Theorem 2.1, where F: D ___, 2Y is defined by F(x) = {f(x)} for all x E D.
In this case F(x) + K is closed for every x E D, since the cone K is closed.
I
REMARK 2.3. The compactness ofT is essential. Indeed, consider X = Y =~and K = ~+'
and let f : D = [0, 1] ___, ~be defined by f(x) = 1 if x E [0, 1/2], and f(x) = 0 if x E ]1/2, 1].
Then f is both quasiconvex and weakly convex with respect toT= ]0, 1[, but f is not convex.
REMARK 2.4. Corollary 2.2 generalizes some known characterization theorems given for realvalued convex functions, such as
(a) Proposition 3 in [1], where X= ~n, Y = ~' K = ~+' D c ~n is a nonempty convex open
set, and T = {1/2};
(b) Theorem 2 in [2], where X is a linear space, Y = ~ , K = ~+, D C X is a nonempty
convex set, and T = {1/2};
(c) Theorem 3 in [3], where X = ~n, Y = ~' K = JR+, D C lRn is a nonempty convex set,
and T ={a} with a E]O, 1[.
REFERENCES
1. K. Nikodem, On some class of midconvex functions, Ann. Polon. Math. 50 (2), 145-151, (1989).
2. F.A. Behringer, Convexity is equivalent to midpoint convexity combined with strict quasiconvexity, Optimization 24 (3/4), 219-228, (1992).
3. X.M. Yang, K.L. Teo and X.Q. Yang, A characterization of convex function, Appl. Math. Lett. 13 (1), 27-30,
(2000).
4. A.W. Roberts and D.E. Varberg, Convex Functions, Academic Press, New York, (1973).
5. W.W. Breckner and G. Kassay, A systematization of convexity concepts for sets and functions, J. Convex
Anal. 4 (1), 109-127, (1997).
6. A. Aleman, On some generalizations of convex sets and convex functions, Anal. Numer. Theor. Approx. 14
(1), 1-6, (1985).
7. H. Hartwig, Generalized convexities of lower semicontinuous functions, Optimization 16 (5), 663-668, (1985).
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