A DUAL GENERALIZATION OF CONVEX FUNCTIONS

. As it is well known, the convexity property of a function may be described by the quasiconvexity property of all “the dual perturbations” of this function. If we consider the “dual perturbation” only in a subset M ⊂ X ∗ we obtain a general class of functions called M -convex. In this paper we establish some special properties and a continuity theorem of this new type of functions.

Taking as starting point the Crouzeix characterization of convex function by quasiconvexity property of all "dual perturbation" (see [7]), in an earlier paper we introduced a new type of convexity, only the "dual perturbation" in a given subset M ⊂ X * .
In the sequel, X denotes a real linear normed space and X * its topological dual.The symbol (•, •) will be used for the usual pairing between X and X * , while •, • will be used for the associated bilinear functional, i.e. x, x * = x * (x), for all x ∈ X, x * ∈ X * .
When Dom(f ) is nonempty we say that f is proper.
A function f : X −→ R ∪ {+∞} is said to be convex if for every x 1 , x 2 ∈ X and for every λ ∈ [0, 1] we have: A function f is called quasiconvex if for every x 1 , x 2 ∈ X and for every λ ∈ [0, 1] we have: Also, it is well known that a function is quasiconvex if and only if its level sets L(f, α) = {x ∈ X| f (x) ≤ α} are convex for every α ∈ R.
Now, we remind the definition of M -convex functions, introduced in [1].
If M is a nonempty subset of X * , we say that the function f : are convex for every α ∈ R.
If −f is M -convex we say that f is M -concave.Throughout this paper, for a given nonempty subset M ⊂ X * , we will denote by C(M ) the class of all M -convex functions.From this definition, we observe that if the set M contains the origin then we obtain a new type of convexity which lies between quasiconvexity and convexity.
For the beginning, we recall some property of this functions proved in [1].
Similarly to the convex case ( [4], [5], [10]), the M -convex functions can be characterized with the aid of its one dimensional restrictions.
In the following result, proved in [1], we characterize the M -convex functions using only its values on the line segments, establishing a characteristic inequality of convex type.
x * , x − y , for every x, y ∈ X, and λ ∈ [0, 1] , where In fact, by (2) we observe that inf On the other hand, if we have is fulfilled for all x, y ∈ X, then f is constant.We recall that the support functional of a set A ⊂ X * , σ A is defined by Thus, the inequality (1) can be rewritten as Now, we consider some special cases for the set M .Thus, if M is a convex set, we have that (4) inf Considering this, it is easy to prove that if f : X −→ R∪{+∞} is a Lipschitz function with constant L and f is S * (0, L)-convex, then f is convex.
In [12], H. X. Phu and P. T. An introduce the notion of s-quasiconvex (s from "stable") functions, and he show that this functions are stable with respect to the following properties: "all lower level sets are convex", "each local minimum is a global minimum", "each stationary point is a global minimizer".
We specify that a function f : The authors show that this functions can be characterized as follows a function f is s-quasiconvex if and only if there exists ε > 0 such (5) that f − x * is quasiconvex, for every x * with the property x * < ε.
Remark 4. The relation ( 5) can be obtained (on the other way) from Theorem 3, taking M a ball with the center in the origin.In fact, as we see in the relation (5), the class of s-quasiconvex functions are the same with the class of M -convex functions, with 0 ∈ int(M ).Now we will consider the sets M ⊆ X * with the following property (P ) for every x ∈ X\{0}, there exists a sequence It is easy to prove that if 0 ∈ int(M ), then the set M has the property (P ).
It is easily to prove that the set M has the property (P ).The function defined by In the following proposition we present a sufficient condition from property (P ).Proposition 6.Let X be a normed linear spaces such that dim(X) ≥ 2 and M ⊆ X * a bounded set, with the property con(M ) w * = X * .Then the set M has the property (P ).
Proof.We start with x ∈ X\{0}.Since dim(X) ≥ 2 then there exists x * , y * ∈ X * \{0} such that x * , x = 0 and y * , x > 0. For every n ∈ N we consider y Taking into account that con(M ) n and λ n y * n ∈ M for every n ∈ N. To prove that the set M has the property (P ) we passing to limit in the following relation and we obtain that λ n y * n , x 0, because M is bounded.
Lemma 7. Let X be a linear normed space and f : Using the characterization given by Theorem 2, we obtain that f is S * (0; r)convex.
Proof.The theorem is a consequence of Lemma 7 and the characterization of s-quasiconvex functions.
Theorem 9. Let M ⊆ X * be a set such that 0 ∈ int(M ).If f is M -convex then for every x, y ∈ X, with f (x) ≤ f (y), there exists α ∈ (0, 1] such that: and the proof is complete.
When M ⊂ X * is a cone, we denote Since M is a cone, then inf and taking into account that f is a M -convex function we obtain that We proceed similarly when y − x ∈ M ♦ .This proved that f is a convex function on the set A.
Now, let us consider the special case of linear subspaces of X * .
Theorem 11.Let M be a proper linear subspace of X * and let f : X −→ R ∪ {+∞} be a M -convex function.Then: Proof.Since M is a linear subspace then the properties (i) and (ii) follow immediately by Proposition 1 (properties (i) and (v)).(iii) Let Y be a linear subspace such that Y ∩ M ⊥ = {0}.Let us take x, y ∈ Y, Since M is a proper linear subspace there exists x * ∈ M such that x * , x − y = f (x) − f (y), and so inf Corollary 12. Let M be a subset of X * such that spanM = X * and λM ⊆ M for every λ ≥ 0. Then every M -convex function is convex.
Proof.If spanM = X * then M ⊥ = {0}.Following the proof of above theorem (iii), we observe that if λM ⊆ M for every λ ≥ 0, then f |Y is convex, whenever Y is a linear subspace.Taking Y = X, we obtain that f is a convex function.
In Proposition 1 (iv), we see that if M w * = X * then, every M -convex function is also a convex function.But this sufficient condition is not necessary.In an earlier paper we established one more general result concerning the equality for every x * ∈ M 2 , and x ∈ X\ {0} , there exists a sequence Taking now M 2 = X * the property ( 6) can be written in the following form (7) for every α ∈ R, and Consequently, we obtain a characterization of the special cases when Mconvexity coincides with convexity.
It is easily to prove that if M w * = X * , then the set M has the property ( 7), but conversely is not always true, as we can see if we consider X = l 1 and

THE EXTREME POINTS OF M -CONVEX FUNCTIONS
In the sequel we shall be concerned with a family of functions that lies between the family of strictly quasiconvex functions and the family of the semistrictly quasiconvex functions.
Let us recall that a function f is strictly quasiconvex [4] if for every x = y, and 0 < λ < 1.
An important property of the convex functions is that every local minimum is a global one.This property, however, holds for more general families of functions (for instance, the family of semistrict quasiconvex functions, see [4]).In this line we consider the sets M ⊆ X * which satisfy the property (P ).
Considering a set M with the property (P ), we want to see the relationships between the family of M -convex functions and the families of generalized convex functions above defined.
When M ⊂ X * has the property (P ), a M -convex function is not necessary a strictly quasiconvex function as we see if we consider the function f (x) = 1.
The following theorem shows the relationship between M -convexity and semistrict quasiconvexity.
Theorem 14.If M ⊂ X * has the property (P ), then every M -convex function is a semistrictly quasiconvex function.
Corollary 15.If M ⊂ X * has the property (P ) and f ∈ C(M ), then every locally extreme point from Dom(f ) is a minimum global point.Moreover, the set of points at which f attains its global minimum is a convex set.
Proof.If f ∈ C(M ) then by Theorem 14 it follows that f is semistrictly quasiconvex, therefore every locally extreme point from Dom(f ) is a global minimum point (see [4]).If x, y are two global minimum points then inf x * ∈Mx,y x * , x − y = 0, and so, by (1), we obtain that This prove that the set of global minimum points is a convex set.
Remark 16.When M ⊂ X * has the property (P ), the function f may not have the strict local maximum points; moreover, in every locally maximum point the function f is locally constant.If f is M -convex and attains its maximum on int(Dom(f )), then f is constant.When M ⊂ X * has the property (P ), the main difference between semistrictly quasiconvex functions and M -convex functions is that for M -convex functions the set of minimum points is a convex set, property which is not true in the case of semistrictly quasiconvex functions.For example, the function f , defined on R by f (x) = 0 for x = 0 and f (x) = 2 for x = 0, is semistrictly quasiconvex but the set of its global minimum points is not convex.

CONTINUITY OF M -CONVEX FUNCTIONS
In this section we will study the continuity property of M -convex functions.
The following result will be needed later on.
If we suppose that Since M is an open set, there exist α > 0 and x * 0 ∈ X * such that x * 0 , x − y = −1 and x * + αx * 0 ∈ M, for every α ∈ (0, α).Now, taking α sufficiently small, we have From ( 10) and ( 11) we obtain that the set L(f, f (y) − x * , y − , x * ) is not convex, which is not true.Therefore which proved (9).Now we will define a special type of radial convexity.
Definition 18.We say that the function f : X → R ∪ {+∞} is radial upper convex if for every x ∈ X, there exists ε x > 0 such that for every v ∈ X, with v = 1, the function is radial upper convex but is not convex.Is obvious that this function is not convex.To prove that she is radial upper convex, we show that for every x, v ∈ X, with v = 1, the function F : R −→ R, defined by is quasiconvex for every α ∈ (−∞, −1) ∪ (1, +∞).Since F is continuous, is sufficiently to prove that ∂F is an quasimonotone operator for every α ∈ (−∞, −1) ∪ (1, +∞).Taking into account that | v, J(x + tv | ≤ 1 for every t with the property x + tv ≤ 1, we obtain that ∂F is quasimonotone (see [2], [5]).We denote by J the duality mapping between X and X * , defined by: It is well known (see [5]) that this mapping is the subdifferential of the function In the sequel, we say that the set M ⊆ X * has property (P * ) if satisfy (P * ) for every x ∈ X\{0} exist {x * n } n∈N ⊆ M such that x * n , x → ∞.Theorem 20.Let M be a set with property (P * ) and let f : X → R∪{+∞} be a M -convex function.Then f is radial continuous on ri(dom f ).
Proof.Without loose our generality, considering Theorem 2, we can suppose that f is defined on R, otherwise we take F (t) = f (x + tv) for x, v ∈ X.Thus, we must to prove that f is continuous.If x 0 ∈ ri(dom f ) and we suppose that f is not lower semicontinuous in x 0 , then there exist > 0 and a net {x n } n∈N with x n → x 0 such that (12) − > f (x n ) − f (x 0 ), for every n ∈ N.
We can suppose that x n > x 0 , for every n ∈ N (analogue when x n < x 0 ).Since x 0 ∈ ri(dom f ), there exists u 1 < x 0 such that u 1 ∈ dom f.Taking into account that f is M -convex and the property (P * ) is fulfilled, then there exists x * ∈ M such that But, by (12), exists u 2 > x 0 such that Considering ( 13) and ( 14) we obtain that L(f, f (x 0 ) − 2 , x * ) is not convex i.e. f is not M -convex.Therefore f must be lower semicontinuous in x 0 .Now, if we suppose, by a contradiction, that f is not upper semicontinuous in x 0 ∈ ri(domf ), then there exist a net {x n } n∈N ⊂ dom(f ) and > 0 such that x n → x 0 and f (x n ) − f (x 0 ) > , for every n ∈ N.
Following the same steps as in previous case we obtain that there exist v 1 , v 2 ∈ dom(f ) and and ( 16) But, by ( 15) and ( 16) we find that L(f, f (x 0 ) + 2 , x * ) is not convex.This ended the proof.
Remark 21.In the above theorem the condition that f to be M -convex can be replaced by the following property: for every x ∈ X and v ∈ X there exist Particularly, this property holds for radial upper convex functions.
Theorem 22.Let f : X → R ∪ {+∞} be a radial upper convex function.If f is bounded from above in a neighborhood of one point x 0 ∈ int(dom f ), then it is locally bounded, that is, for every x ∈ int(dom f ) there exists a neighborhood on which f is bounded.
Proof.We first show that if f is bounded from above in a neighborhood of x 0 , it is also bounded from below in the same neighborhood.Since f is radial upper convex, we can find r x 0 > 0 such that for every v ∈ X with v = 1, the function For every z ∈ S(x 0 ; ε), there exists λ ≥ 0 and y ∈ X such that y = 1 and z = x 0 + λy. Taking which prove that f is bounded from below on S(x 0 , ).Let x ∈ int(dom f ) and x = x 0 .Then x = x 0 + λy, where again y ∈ X, y = 1 and µ is a positive number.
Since x ∈ int(dom f ), there exists α > µ such that v = x 0 + αy ∈ int(domf ).Taking λ = µ α , the set , where γ = (1 − λ)ε).Therefore we find r z > 0 such that for every u ∈ V, the function defined by Since f is bounded on S(x 0 ; ε), then which proved that f (u) ≤ K + max{2K, r z r}, for every u ∈ V, i.e. f is also bounded from above on V , as claimed.Now, we establish an extension of well known continuity theorem of convex functions (see [5], [6]).
Theorem 23.Let f : X → R ∪ {+∞} be a radial upper convex function.If f is bounded from above in a neighborhood of one point x 0 ∈ int(dom f ), then f is continuous on int(dom f ).
Proof.By Theorem 22, for each x 0 ∈ int(dom f ) we can find a neighborhood S(x 0 , 2 ) on which f is bounded, i.e. there exist K x 0 > 0 such that |f (x)| ≤ K x 0 for every x ∈ S(x 0 , 2 ).
For the begining we prove that for every x ∈ int(dom f ) we find a constant K x 0 > 0 such that (17) inf Since the function F (t) = f (x 0 +t x−x 0 x−x 0 ) is (−∞, −r x 0 )∪(r x 0 , +∞)-convex, from Lemma 17, we obtain that Therefore f (x 0 ) − f (y) < −2K x 0 and hence f (y) > K x 0 , which is not possible because y ∈ S(x 0 ; 2ε).Consequently, the inequality (17) is always true.Analogue we proof relation (18), using in this case the function F (t) = f (x + t x 0 −x x−x 0 ), which is also a (−∞, −r x 0 ) ∪ (r x 0 , +∞)-convex function.Now, if we suppose that f is not continuous in x 0 , then we find > 0 and a sequence {x n } n∈N ⊆ S(x 0 ; ε) such that x n → x 0 , From ( 19) and (20) we find n 0 ∈ N such that for every n ≥ n 0 , From above relations we obtain that |f (x n ) − f (x 0 )| ≤ K x 0 x n − x 0 , for every n ≥ n 0 , which is not possible because is a contradiction with (19) or (20).
Remark 24.It is obvious that in the special case M = X * we obtain the usual continuity theorem of convex functions.