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Outer Γ-Convexity in Vector SpacesHoang Xuan Phu aa Institute of Mathematics, Vietnamese Academy of Science andTechnology , Hanoi, VietnamPublished online: 17 Sep 2008.

To cite this article: Hoang Xuan Phu (2008) Outer Γ-Convexity in Vector Spaces, Numerical FunctionalAnalysis and Optimization, 29:7-8, 835-854, DOI: 10.1080/01630560802282250

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Numerical Functional Analysis and Optimization, 29(7–8):835–854, 2008Copyright © Taylor & Francis Group, LLCISSN: 0163-0563 print/1532-2467 onlineDOI: 10.1080/01630560802282250

OUTER � -CONVEXITY IN VECTOR SPACES

Hoang Xuan Phu

Institute of Mathematics, Vietnamese Academy of Science and Technology,Hanoi, Vietnam

� A subset S of some vector space X is said to be outer � -convex w.r.t. some givenbalanced subset � ⊂ X if for all x0, x1 ∈ S there exists a closed subset � ⊂ [0, 1] such that�x� | � ∈ �� ⊂ S and [x0, x1] ⊂ �x� | � ∈ �� + 0�5 � , where x� := (1 − �)x0 + �x1. A real-valued function f : D → � defined on some convex D ⊂ X is called outer � -convex if forall x0, x1 ∈ D there exists a closed subset � ⊂ [0, 1] such that [x0, x1] ⊂ �x� | � ∈ �� + 0�5 �and f (x�) ≤ (1 − �)f (x0) + �f (x1) holds for all � ∈ �. Outer � -convex functions possesssome similar optimization properties as these of convex functions, e.g., lower level sets of outer� -convex functions are outer � -convex and � -local minimizers are global minimizers. Someproperties of outer � -convex sets and functions are presented, among others a simplex propertyof outer � -convex sets, which is applied for establishing a separation theorem and for provingthe existence of modified subgradients of outer � -convex functions.

Keywords Generalized convexity; Global minimizer; � -local minimizer; Nonconvexitymeasure; Outer � -convexity; Rough convexity; Self-Jung constant; Separationtheorem; Simplex property; Subgradients of nonconvex functions.

AMS Subject Classification 52A01; 52A30; 52A41; 49K27; 90C26.

1. INTRODUCTION

Throughout this paper, X denotes a vector space over the field ofreal numbers, which is sometimes specified as topological vector space ornormed vector space, and � is a balanced subset of X , i.e.,

�� ⊂ � for every � ∈ � with |�| ≤ 1,

Address correspondence to Hoang Xuan Phu, Institute of Mathematics, Vietnamese Academyof Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam; E-mail: hxphu@math.ac.vn

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which is sometimes assumed in addition to have the origin 0 of X as aninternal point, i.e.,

∀x ∈ X ∃� > 0 : |�| ≤ � ⇒ �x ∈ ��

For any x0 and x1 in X , the following notations are used

x� := (1 − �)x0 + �x1,

[x0, x1] := �x� | 0 ≤ � ≤ 1�,

]x0, x1[ := �x� | 0 < � < 1��

S ⊂ X is said to be an outer � -convex set if for all x0, x1 ∈ S there exists aclosed subset � of interval [0, 1] such that

�x� | � ∈ �� ⊂ S , [x0, x1] ⊂ �x� | � ∈ �� + 0�5 �� (1.1)

Note that only x0, x1 ∈ S with x0 − x1 �∈ � have to be considered, becausex0 − x1 ∈ � always yields [x0, x1] ⊂ �x0, x1� + 0�5 � , i.e., (1.1) is automaticallyfulfilled for � = �0, 1�.

Obviously, if (1.1) is satisfied for some closed � ⊂ [0, 1], then it followsimmediately that

[x0, x1] ⊂ ([x0, x1] ∩ S) + 0�5 � , (1.2)

but not vice versa. For instance, for X = �, S = �0�∪ ]1, 2], � = [−1, 1],x0 = 0, and x1 = 2, we have

[x0, x1] = [0, 0�5]∪]0�5, 2] ⊂ (�0�∪]1, 2]) + [−0�5, 0�5] = ([x0, x1] ∩ S)+0�5 � ,

but there is no closed � ⊂ [0, 1] such that (1.1) is fulfilled.Let D be a convex subset of X . A real-valued function f : D → � is

said to be an outer � -convex function if for all x0, x1 ∈ D there exists a closedsubset � ⊂ [0, 1] such that

[x0, x1] ⊂ �x� | � ∈ �� + 0�5 � (1.3)

and

∀� ∈ � : f (x�) ≤ (1 − �)f (x0) + �f (x1)� (1.4)

In particular, if � has the origin 0 ∈ X as the only element, i.e., � = �0�,then (1.1) is equivalent to [x0, x1] ⊂ S and � = [0, 1], and (1.3)–(1.4) areequivalent to

∀� ∈ � = [0, 1] : f (x�) ≤ (1 − �)f (x0) + �f (x1)�

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Outer � -Convexity in Vector Spaces 837

That means that the outer � -convexity w.r.t. � = �0� is just the ordinaryconvexity.

If X is a normed vector space,

�B(x , r ) := �x ′ ∈ X | ‖x − x ′‖ ≤ r �, x ∈ X , r > 0,

and is a positive real number, then the outer � -convexity w.r.t. � = �B(0, )is identical with the outer -convexity introduced in Phu and An [1].Hence, we continue to write outer -convexity for the outer � -convexityw.r.t. � = �B(0, ).

Other kinds of roughly convex functions, such as global �-convexityintroduced by Hu et al. [2], rough -convexity proposed by Klötzlerand investigated by Hartwig [3], -convexity defined in Phu [4, 5], andsymmetrical -convexity in Hai and Phu [6], are special cases of the outer-convexity studied in Phu and An [1], hence they do belong to the outer� -convexity, too.

One of the main aims for the mentioned rough generalizations isto get a wide class of nonconvex functions that are somehow roughlyconvex in order to still possess some similar optimization properties asof convex functions, e.g., lower level sets of roughly convex functionsare roughly convex and local minimizers of roughly convex functions areglobal minimizers, where the notion of local minimizers must be modifiedaccordingly.

The outer � -convexity introduced in this paper is not only a puregeneralization for more general spaces, but also it is an essential extension,even for normed vector spaces. It allows us to work with nonconvexand/or unbounded � , which is not the case when � = �B(0, ). Using thisadvantage, we can describe partially convex sets and functions as presentedin another paper and deal with the epigraph to obtain different results asdone in Theorems 3.5 and 3.8 of this paper.

Some properties of outer � -convex sets are presented in Section 2.The crucial result is a simplex property saying that, for an outer � -convexand relatively closed subset S , each point z∗ ∈ conv S\S belongs to somesimplex conv�z∗

1 , z∗2 , � � � , z

∗k � whose vertices z∗

i , 1 ≤ i ≤ k, lie in S and satisfyz∗i − z∗

j ∈ � for 1 ≤ i , j ≤ k (Theorem 2.4). This yields an estimation fornonconvexity measure of outer -convex sets in normed vector space Xby conv S ⊂ S + �B(0, Js(X )/2), where Js(X ) is the self-Jung constant of X(Theorem 2.9). A further consequence is a separation property of outer-convex sets saying that if �B(x , r ) ∩ S = ∅ for some r > 1

2 Js(X ), thenx �∈ cl(conv S), and therefore, some non-zero continuous linear functionalstrictly separates x and S (Theorem 2.10).

Section 3 is devoted to some properties of outer � -convex functions.As expected, all lower level sets of an outer � -convex function are outer� -convex, and f : D ⊂ X → � is outer � -convex w.r.t. � = B if and only

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if its epigraph epi f is outer � -convex w.r.t. � = B × � (Theorem 3.5).Theorems 3.6 and 3.7 deal with the optimization property that � -localminimizers of an outer � -convex function are global minimizers. A specialapplication of the simplex property stated in Theorem 2.4 is Theorem 3.8on the existence of a modified subgradient of outer � -convex functions.For instance, if D is some compact convex subset of a finite-dimensionalnormed vector space X and if f : D → � is bounded from below, lowersemicontinuous, and outer � -convex w.r.t. � = �B(0, ), then for all z∗ ∈ riD,there is � ∈ �n such that

∃z ∈ �B(z∗, Js(X )/2) ∀z ∈ D : f (z) ≥ f (z) + 〈�, z − z〉�

2. OUTER � -CONVEX SETS

Let us begin with some statements whose proof is obvious and there-fore omitted.

Proposition 2.1.

(a) Every convex set is outer � -convex w.r.t. arbitrary balanced � .(b) The intersection of a convex set and an outer � -convex set is outer � -convex.(c) If a set is outer � -convex w.r.t. � = B, then it is outer � -convex w.r.t. every

larger balanced set � = B ′ ⊃ B�

The class of outer � -convex sets is very large. A simple example is givenin the following.

Example 2.2. Let S be the set of all points in �n whose coordinates arerational numbers and let � be any balanced set in �n having the origin 0 asan internal point. Then S is outer � -convex. Indeed, for arbitrary x0, x1 ∈ Swith x0 − x1 �∈ � , just take any rational � > 0 satisfying �(x0 − x1) ∈ � and

� = �0, 1� ∪ �� ∈ ]0, 1[ | � = n�, n ∈ ��,

then this � is closed, �x� | � ∈ �� ⊂ S , and [x0, x1] ⊂ �x� | � ∈ �� + 0�5 � ,i.e., (1.1) is satisfied. By Proposition 2.1, for any convex set C ⊂ �n , C ∩ Sis outer -convex w.r.t. arbitrary balanced set � ⊂ �n having the origin asan internal point.

Other examples can be easily derived from the next assertion.

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Outer � -Convexity in Vector Spaces 839

Proposition 2.3. Let X be a normed vector space, C ⊂ X be convex, S� ⊂ Xand diam(S�) < for all � from an arbitrary index set A. Assume

cl(S�) ∩ cl( ⋃

�′∈A\���S�′

)= ∅ for all � ∈ A� (2.1)

Then C\ ⋃�∈A S� is outer � -convex w.r.t. � = �B(0, ).

Proof. Assume the contrary that S = C\ ⋃�∈A S� is not outer � -convex

w.r.t. � = �B(0, ). Then there exist x0, x1 ∈ S such that (1.1) does not holdtrue for any closed � ⊂ [0, 1]. It follows that there are x ′

0 and x ′1 such that

[x ′0, x

′1] ⊂ [x0, x1]\S = [x0, x1] ∩

⋃�∈A

S� and ‖x ′0 − x ′

1‖ ≥ �

Let �∗ ∈ A be a certain index with [x ′0, x

′1] ∩ S�∗ �= ∅. Then [x ′

0, x′1] cannot

be completely contained in S�∗ , otherwise diam(S�∗) ≥ ‖x ′0 − x ′

1‖ ≥ , whichconflicts with the assumption diam(S�) < for all � ∈ A. Therefore,[x ′

0, x′1] ∩ S�∗ has at least a boundary point x∗ (relative to the line through

x ′0 and x ′

1), which is also a cluster point of the complement

[x ′0, x

′1]\

([x ′0, x

′1] ∩ S�∗

) = [x ′0, x

′1]\S�∗ = [x ′

0, x′1] ∩

⋃�′∈A\��∗�

S�′ ,

i.e., x∗ belongs to the closure of⋃

�′∈A\��∗� S�′ , which conflicts with (2.1).Hence, S = C\ ⋃

�∈A S� must be outer � -convex w.r.t. � = �B(0, ). �

Roughly speaking, Proposition 2.3 says that a piece of cheese is roughlyconvex, i.e., outer � -convex w.r.t. � = �B(0, ), provided that the diametersof cheese holes are less than .

Note that (2.1) cannot be replaced by pairwise disjoint conditioncl(S�) ∩ cl(S ′

�) = ∅ for � �= �′. For instance, if all S�, � ∈ A, are singleton,then they are all closed, pairwise disjoint, and all of their diameters arezero. But the union

⋃�∈A S� can create any set, for which C\ ⋃

�∈A S� is notouter � -convex.

Let us come to the crucial result of this section.

Theorem 2.4. Let X be a locally convex topological vector space. Let S ⊂ X beouter � -convex and closed relative to each convex hull of finite points from S. Ifz∗ ∈ conv S\S, then there exist k ≥ 2 and z∗

i ∈ S, i = 1, 2, � � � , k, such that

z∗ ∈ conv�z∗1 , z

∗2 , � � � , z

∗k � (2.2)

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and

z∗i − z∗

j ∈ � , 1 ≤ i , j ≤ k, (2.3)

i.e., [z∗i , z

∗j ] ⊂ �z∗

i , z∗j � + 0�5 � for 1 ≤ i , j ≤ k. In particular, if dimX = n then

k ≤ n + 1.

Proof. (a) For z∗ ∈ conv S , there are k ≥ 2 and z0i ∈ S , 1 ≤ i ≤ k, suchthat

z∗ ∈ Z 0 := conv�z0i | 1 ≤ i ≤ k��

In particular, if dimX = n, by Caratheodory’s theorem we can assumek ≤ n + 1. Because X is a topological vector space, Z 0 is compact.

(b) Consider

F := �Z∣∣ z∗ ∈ Z = conv�z1, z2, � � � , zk�, zi ∈ Z 0 ∩ S for 1 ≤ i ≤ k��

This system is partially ordered by inclusion, i.e., Z ′ ≤ Z ′′ ⇔ Z ′ ⊇ Z ′′.

(c) Assume that F ′ is a totally ordered subsystem of F . We are goingto show that F ′ has an upper bound �Z ∈ F , i.e., there exist zi ∈ Z 0 ∩ S , 1 ≤i ≤ k, such that

z∗ ∈ �Z = conv�z1, z2, � � � , zk� and Z ≤ �Z for all Z ∈ F ′� (2.4)

Z 0 ∩ S is compact, because it is closed and Z 0 is compact.Consequently,

P Z 0 := (Z 0 ∩ S)k = (Z 0 ∩ S) × · · · × (Z 0 ∩ S)︸ ︷︷ ︸k times

is compact in its product topology (Tychonoff theorem).Let us denote

yZ := (z1, z2, � � � , zk) if Z = conv�z1, z2, � � � , zk� ∈ F �

Because (F ′,≤) is totally ordered, every finite subset of F ′ has the smallestsimplex as an upper bound, i.e., (F ′,≤) is directed. Hence, (yZ )Z∈F ′ is ageneralized sequence in the compact space P Z 0

, and therefore, (yZ )Z∈F ′ hasa cluster point y = (z1, z2, � � � , zk) ∈ P Z 0

(see Lemma I.7.9 in Dunford andSchwartz [7, p. 26]). Because zi ∈ Z ∩ S for 1 ≤ i ≤ k and Z ∈ F ′, we have

yZ ∈ P Z := (Z ∩ S)k �

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Outer � -Convexity in Vector Spaces 841

Clearly, for every Z ′ ∈ F ′, P Z ′ ⊇ P Z if Z ′ ≤ Z , which implies that thesubsequence (yZ )Z∈F ′,Z ′≤Z is a generalized sequence in the compact set P Z ′

.Of course, y is a cluster point of this subsequence, too. It follows that y ∈P Z ′

for every Z ′ ∈ F ′ (see Lemma I.7.2 in Dunford and Schwartz [7, p. 27]),i.e.,

zi ∈ Z ′ ∩ S ⊆ Z 0 ∩ S for 1 ≤ i ≤ k, Z ′ ∈ F ′,

which yields

Z ′ ≤ �Z = conv�z1, z2, � � � , zk� for all Z ′ ∈ F ′�

For (2.4), it remains to show that z∗ ∈ �Z . Assume the contrary thatz∗ �∈ �Z . For X is a locally convex linear topological space, there is abalanced and convex neighborhood U of 0 with (�z∗� + U ) ∩ �Z = ∅, whichis equivalent to

z∗ �∈ �Z + U � (2.5)

Because y is a cluster point, there is at least a Z ∈ F ′ such that yZ iscontained in �y� + U k , which implies i := zi − zi ∈ U for 1 ≤ i ≤ k. z∗ ∈ Zimplies that there are �i ≥ 0 with

∑�i = 1 and

z∗ =k∑

i=1

�i zi =k∑

i=1

�i(zi + i) =k∑

i=1

�i zi +k∑

i=1

�i i ∈ �Z + U ,

in contradiction to (2.5). Hence, z∗ ∈ �Z ∈ F , and �Z is an upper bound ofthe totally ordered subsystem F ′.

(d) Because every totally ordered subsystem F ′ of F has an upperbound �Z ∈ F , Zorn’s lemma yields that the partially ordered system F hasa maximal element Z ∗ with

z∗ ∈ Z ∗ = conv�z∗1 , z

∗2 , � � � , z

∗k �, z

∗i ∈ Z0 ∩ S for 1 ≤ i ≤ k,

i.e., (2.2) holds true.

(e) Assume the contrary that (2.3) does not hold, for instance z∗1 −

z∗2 �∈ � . Because S is outer � -convex, there is z ∈ ]z∗

1 , z∗2 [ ∩S . For

Z ∗1 := conv�z∗

1 , z, z∗3 , � � � , z

∗k � and Z ∗

2 := conv�z, z∗2 , z

∗3 , � � � , z

∗k �

we have Z ∗ = Z ∗1 ∪ Z ∗

2 . Therefore, one of them must contain z∗, forinstance z∗ ∈ Z ∗

1 . Clearly, Z∗1 ∈ F and Z ∗ ≤ Z ∗

1 and Z ∗ �= Z ∗1 , which conflicts

with the maximality of Z ∗. Hence, (2.3) is satisfied, too. �

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In the above proof, part (c) for proving the existence of an upperbound of a totally ordered subsystem is the longest one. That seems tobe unsuitable, because by finite intersection property of compact sets, it isobviously that

⋂Z∈F ′ Z is nonempty. But the point is that this intersection

is not necessarily the convex hull of k points from Z0 ∩ S , which containsz∗, in order to be an element of F .

Corollary 2.5. Assume that � is convex and S ⊂ X is outer � -convex and closedrelative to each convex hull of finite points from S. Then conv S ⊂ S + � .

Proof. Due to Theorem 2.4, for arbitrary z∗ ∈ conv S\S , there exist k ≥ 2and z∗

i ∈ S , 1 ≤ i ≤ k, such that z∗ ∈ conv�z∗1 , z

∗2 , � � � , z

∗k � and z∗

i − z∗j ∈ � for

1 ≤ i , j ≤ k. Following, z∗i ∈ �z∗

1� + � for 2 ≤ i ≤ k. Because � is convex, wehave then

z∗ ∈ conv�z∗1 , z

∗2 , � � � , z

∗k � ∈ �z∗

1� + � ⊂ S + ��

Hence, conv S ⊂ S + � . �

Note that the convexity assumption of � in Corollary 2.5 is reallyneeded, as the following shows.

Example 2.6. Let S = �(0, 0), (0, 1), (1, 0)� ⊂ X = �2 and

� = �(0, t) | −1 ≤ t ≤ 1� ∪ �(t , 0) | −1 ≤ t ≤ 1� ∪ �(t ,−t) | −√2 ≤ t ≤ √

2��

Then S is outer � -convex, but for all �1 > 0, �2 > 0, �3 > 0 satisfying�1 + �2 + �3 = 1 and � > 0, we always have

�1(0, 0) + �2(0, 1) + �3(1, 0) �∈ S + �� ,

i.e., conv S �⊂ S + � .

For normed vector spaces and � = �B(0, ) = �x ∈ X | ‖x‖ ≤ �,Theorem 2.4 implies immediately the following.

Theorem 2.7. Let X be a normed vector space and S ⊂ X be outer -convexw.r.t. some given > 0 and closed relative to each convex hull of finite pointsfrom S. If z ∈ conv S\S, then there exist zi ∈ S, i = 1, 2, � � � , k, such that z ∈conv�z1, z2, � � � , zk� and

‖zi − zj‖ ≤ for 1 ≤ i , j ≤ k�

In particular, if dimX = n then k ≤ n + 1.

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Outer � -Convexity in Vector Spaces 843

Note that this theorem was proved in Phu [8], but there was an errorin its proof.

Corollary 2.8. Let S ⊂ X be an outer -convex subset of some normed space X(which is not necessarily closed). If z ∈ conv S\S, then there exist zi ∈ cl S, i =1, � � � , k, such that z ∈ conv�z1, z2, � � � , zk� and

‖zi − zj‖ ≤ for 1 ≤ i , j ≤ k�

Proof. Clearly, we only need to consider z �∈ cl S , i.e.,

z ∈ conv S\cl S ⊂ conv(cl S)\cl S �By Proposition 2.4 in Phu and An [1], cl S is outer -convex, too.Therefore, the assertion follows immediately from Theorem 2.7 appliedto cl S . �

Let

Js(X ) := sup{2rconv S(S)diam S

: S ⊂ X bounded, non-empty, non-singleton},

where rconv S(S) = infx∈conv S

supy∈S

‖x − y‖, diam S = supx ,y∈S

‖x − y‖,

be the so-called self-Jung constant of some normed vector space X . Let S ⊂ Xbe bounded and z ∈ conv S\S . Then, by Proposition 3.1 in Phu [9], we have

∃s ∈ S : ‖z − s‖ ≤ 12Js(X )diam S �

Applying this fact and Theorem 2.7, we showed the following estimationof nonconvexity measure.

Theorem 2.9 [8]. Let S be a closed and outer -convex subset of some normedvector space X . Then

∀y ∈ conv S\S ∃z ∈ S : ‖y − z‖ ≤ 12Js(X ),

i.e., conv S ⊂ S + �B(0, r ) if r ≥ 12 Js(X ).

By using the preceding result, we derived the following separationtheorem, where S is not necessarily closed.

Theorem 2.10 [8]. Let S be an outer -convex subset of some normed vectorspace X . If �B(x , r ) ∩ S = ∅ for some r > 1

2 Js(X ), then x �∈ cl(conv S), andtherefore, some non-zero continuous linear functional strictly separates x and S.

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Note that, if X is finite-dimensional and S is compact, x �∈ conv Sfollows from �B(x , r ) ∩ S = ∅ even for r = 1

2 Js(X ). Hence, some non-zerocontinuous linear functional strictly separates x and S in this case, too.

To apply Theorems 2.9 and 2.10 for concrete normed vector spaces, itremains to insert corresponding values of self-Jung constant Js(X ). Some ofthem are as follows. Based on pioneering research of Jung [10] and furtherinvestigations by Klee [11], Maluta [12], and Routledge [13], it holds forn-dimensional Euclidean space �n and for the infinite-dimensional Hilbertspace �2

Js(�n) = (2n/(n + 1)

)1/2and Js(�2) = √

2�

Amir [14] showed

Js(X ) ≤ 2nn + 1

if dimX = n�

Due to Pichugov [15],

Js(�p) = Js(Lp[0, 1]) = max�21/p , 21−1/p�, 1 ≤ p < ∞�

Self-Jung constant for further spaces can be found in Maluta [12], forinstance

Js(X ) = 1 if X = (�2, ‖ · ‖∞),

Js(X ) = 2 if X is a non-reflexive Banach space.

Another application of Theorem 2.7 was obtained in Phu [8] by usinga result of Brunn [16], namely:

Proposition 2.11 [8]. Let S be a closed outer -convex set in n-dimensionallinear normed space X . Let S

i be defined recursively as follows:

S 1 :=

⋃x ,y∈S , ‖x−y‖≤

[x , y], S i+1 :=

⋃x ,y∈Si , ‖x−y‖≤

[x , y], i ≥ 1�

Then conv S = S in if in satisfies the inequality 2in−1 ≤ n + 1 ≤ 2in .

In the next section, we will deal with a further application ofTheorem 2.4.

3. OUTER � -CONVEX FUNCTIONS

Throughout this section, we consider outer � -convex functions definedon some convex subset D of vector space X .

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Almost all known kinds of roughly convex functions, such as global �-convexity in Hu et al. [2], rough -convexity in Hartwig [3], -convexityin Phu [4, 5], symmetrical -convexity in Hai and Phu [6], and outer -convexity in Phu and An [1], do belong to the outer � -convexity. Hence,many examples of outer � -convex functions can be read there and also inPhu [17].

An important class of outer � -convex functions is given in Phu [18].A further example is as follows.

Example 3.1. Let zi ∈ �, i ∈ �, satisfying

0 < � < zi+1 − zi ≤ , i ∈ �, (3.1)

be given and g : � → � be any function satisfying

g (x) ≥ g (zi) = 0 for all x ∈ �, i ∈ �� (3.2)

Then g is outer -convex, i.e., outer � -convex w.r.t. � = �B(0, ), because forarbitrary x0, x1 ∈ �, x0 �= x1, by choosing

�i = (x0 − zi)/(x0 − x1), i ∈ �,

we have x�i = (1 − �i)x0 + �i x1 = zi , which implies

0 < x�i+1 − x�i ≤ , i ∈ �,

and for 0 < �i < 1

g (x�i ) = g (zi) = 0 ≤ (1 − �i)g (x0) + �i g (x1)�

Hence, (1.3) and (1.4) are fulfilled for � = �0, 1� ∪ ��i | 0 < �i < 1�, whichis finite, and therefore, a closed subset of [0, 1].

To get more examples, we can apply the next proposition, whose proofis rather obvious, so it will be omitted.

Proposition 3.2.

(a) Every convex function is outer � -convex w.r.t. arbitrary balanced � .(b) The sum of a convex function and an outer � -convex function is outer

� -convex.(c) The maximum of a convex function and an outer � -convex function is outer

� -convex.(d) If a function is outer � -convex w.r.t. � = B, then it is outer � -convex w.r.t.

every larger balanced set � = B ′ ⊃ B.

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Function g satisfying (3.1)–(3.2) can be considered as one-sideddisturbance with the only condition that in every interval [x , x + ] thereis at least one point where disturbance disappears. By Proposition 3.2, if aconvex function f : � → � is disturbed by such a disturbance g , then thedisturbed function f + g remains at least as an outer -convex one.

An important property of convex functions w.r.t. optimization is thatlower level sets are convex. For outer � -convex functions, we also have thefollowing similar one, whose proof is straightforward and will be omitted.

Proposition 3.3. All lower level sets of an outer � -convex function are outer� -convex.

Proposition 3.4. f : D → � is outer � -convex if and only if for each linearfunctional g : D → �, all level sets of f + g are outer � -convex.

Proof. (a) Necessity: If f is outer � -convex and g is linear, then, byProposition 3.2, f + g is outer � -convex. Therefore, Proposition 3.3 yieldsthat all level sets of f + g are outer � -convex.

(b) Sufficiency: Assume that for each linear functional g : D → �, alllevel sets of f + g are outer � -convex. Consider arbitrary x0, x1 ∈ D. Chooseg such that g (x0 − x1) = f (x1) − f (x0), which yields

� := f (x0) + g (x0) = f (x1) + g (x1)�

Because the level set �x ∈ D | f (x) + g (x) ≤ �� is outer � -convex, by (1.1),there exists a closed subset � ⊂ [0, 1] such that

[x0, x1] ⊂ �x� | � ∈ �� + 0�5 �

and it holds for all � ∈ �

0 ≥ f (x�) + g (x�) − �

= f (x�) + g (x�) − (1 − �)(f (x0) + g (x0)) − �(f (x1) + g (x1))

= f (x�) − (1 − �)f (x0) − �f (x1),

i.e., (1.3)–(1.4) hold. Hence, by definition, f is outer � -convex. �

In order to state an equivalent relation between outer -convex setsand outer -convex functions, the above proposition was proved in Phuand An [1] for normed vector space X and � = �B(0, ). Using lower levelsets for this purpose is rather strange. But the concept of outer -convexsets in normed vector space is not suitable to treat epigraph, because evenfor very small ‖x0 − x1‖, the distance between (x0, y0) and (x1, y1) in X × �

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satisfying f (x0) ≤ y0 and f (x1) ≤ y1 may be extremely large and it may tendto infinity. One of the reasons for introducing the notion of outer � -convexity in this paper is to overcome this hindrance. With an unbounded� , we can establish an equivalence between outer � -convex functions andtheir epigraph as follows.

Theorem 3.5. Let B be a balanced subset of vector space X . Then f : D ⊂ X →� is outer � -convex w.r.t. � = B if and only if its epigraph epi f is outer � -convexw.r.t. � = B × �.

Proof. (a) Necessity: Assume that f is outer � -convex w.r.t. � = B.Consider two arbitrary points (x0, y0) and (x1, y1) in

epi f = �(x , y) ∈ X × � | f (x) ≤ y�

with (x0, y0) − (x1, y1) �∈ B × �, i.e., x0 − x1 �∈ B. By definition in (1.3)–(1.4), there exists a closed subset � ⊂ [0, 1] such that

[x0, x1] ⊂ �x� | � ∈ �� + 0�5B (3.3)

and

∀� ∈ � : f (x�) ≤ (1 − �)f (x0) + �f (x1)�

Because f (x0) ≤ y0, f (x1) ≤ y1, and � ∈ [0, 1], it follows that∀� ∈ � : f (x�) ≤ (1 − �)y0 + �y1,

i.e.,

∀� ∈ � : (1 − �)(x0, y0) + �(x1, y1) = (x�, (1 − �)y0 + �y1) ∈ epi f �

On the other hand, (3.3) implies immediately

[(x0, y0), (x1, y1)] ⊂ �(1 − �)(x0, y0) + �(x1, y1) | � ∈ �� + 0�5B × ��

Hence, (1.1) holds for S = epi f and � = B × �, and therefore, epi f isouter � -convex w.r.t. � = B × �.

(b) Sufficiency: Assume that epi f is outer � -convex w.r.t. � = B × �.Consider two arbitrary points x0 and x1 in D. Because (x0, f (x0)) and(x1, f (x1)) belong to epi f , due to (1.1), there exists a closed subset � ⊂[0, 1] such that

[(x0, f (x0)), (x1, f (x1))] ⊂ �(1 − �)(x0, f (x0))

+�(x1, f (x1)) | � ∈ �� + 0�5B × � (3.4)

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and

∀� ∈ � : (1 − �)(x0, f (x0)) + �(x1, f (x1)) ∈ epi f � (3.5)

Equation (3.4) is equivalent to

[x0, x1] ⊂ �(1 − �)x0 + �x1 | � ∈ �� + 0�5B

and Equation (3.5) is equivalent to

∀� ∈ � : f ((1 − �)x0 + �x1) ≤ (1 − �)f (x0) + �f (x1),

therefore, (1.3) and (1.4) hold true for � = B. Hence, f is outer � -convexw.r.t. � = B. �

Thus by using the unbounded component � in � = B × �, we canwork with all points in epigraph independently of the difference offunction values.

Another important property of convex functions is that each localminimizer is a global minimizer. One main purpose of the concept of outer� -convexity and of almost all other special kinds of rough convexity is toobtain a similar property for nonconvex functions. Concretely, we have thefollowing assertion.

Theorem 3.6. Suppose that the origin 0 is an internal point of � . Let f : D →� be outer � -convex. Then

f (x∗) = infx∈D∩(�x∗�+�)

f (x) �⇒ f (x∗) = infx∈D

f (x), (3.6)

i.e., if x∗ is a � -minimizer, then it is a global minimizer.

Proof. Assume the contrary that x0 = x∗ is not a global minimizer, thenthere exists x1 ∈ D such that x1 �= x0 and f (x1) < f (x0). Because f is outer� -convex and 0 is an internal point of � , there exists a � ∈ ]0, 1[ such that

x0 �= x� = (1 − �)x0 + �x1, x� − x0 ∈ � , f (x�) ≤ (1 − �)f (x0) + �f (x1)�

It follows that x� ∈ D ∩ (�x∗� + �) and

f (x�) ≤ (1 − �)f (x0) + �f (x1) < f (x0) = f (x∗),

which conflicts with f (x∗) = infx∈D∩(�x∗�+�) f (x). Hence, x∗ must be a globalminimizer of f . �

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Equation (3.6) says that it is only necessary to compare within subsetD ∩ (�x∗� + �) if the origin 0 is an internal point of � . Otherwise, it is notenough. For instance, if � = �0�, then D ∩ (�x∗� + �) = �x∗� and the left-hand part of statement (3.6) is true for all x∗ ∈ D but the right-hand partdoes not hold everywhere. Therefore, if 0 is not an internal point of � , wehave to extend comparison domain. For instance, for a topological vectorspace X , just take a small neighborhood B of the origin and add it to� to obtain larger domain � + B, or it is better (i.e., smaller) to choosethe union � ∪ B. Such a larger domain is used in the following to treat �having the origin 0 not as an internal point.

Theorem 3.7. Let f : D → � be outer � -convex and let � ′ be any subset of Xcontaining � and having the origin as an internal point. Then

f (x∗) = infx∈D∩(�x∗�+� ′)

f (x) �⇒ f (x∗) = infx∈D

f (x),

i.e., a � -local minimizer is a global minimizer.

Proof. Consider an arbitrary x1 ∈ D\� ′. If the origin 0 is not an internalpoint of � ∩ �(1 − �)x∗ + �x1 | � ∈ �� relative to line �(1 − �)x∗ + �x1 | � ∈��, then, by definition, f is convex on this line. Hence, f (x∗) ≤ f (x1)because f (x∗) = infx∈D∩(�x∗�+� ′) f (x) yields that x∗ is a local minimizer of fon the mentioned line. Otherwise, if the origin 0 is an internal point of� ∩ �(1 − �)x∗ + �x1 | � ∈ �� relative to line �(1 − �)x∗ + �x1 | � ∈ ��, thenf (x∗) = infx∈D∩(�x∗�+� ′) f (x) and � ⊂ � ′ imply f (x∗) = infx∈D∩(�x∗�+�) f (x) and,by the same reason as in the proof of Theorem 3.6, we have f (x∗) ≤ f (x1),too. Hence, x∗ is global minimizer of f . �

An important property of convex functions is their subdifferentiability.Recall that a linear functional � ∈ X ∗ is said to be a subgradient of a convexfunction f : D ⊂ X → � at point z∗ ∈ D if

∀z ∈ D : f (z) ≥ f (z∗) + 〈�, z − z∗〉� (3.7)

f is said to be subdifferentiable at z∗ if the subdifferential �f (z∗) containingall subgradients of f at z∗ is not empty.

Assume now that X = �n is equipped with some norm, D ⊂ X isconvex, and let f : D → �, i.e., f is proper and D is the effective domain off . If f is convex, then it is well known that �f (z∗) is nonempty for z∗ fromthe relative interior riD (see Rockafellar [19, p. 217]). If f is not convex,this property does not hold anymore. But we are going to present a similarone for outer � -convex functions. Obviously, (3.7) is equivalent to

∀z ∈ D : f (z) − 〈�, z〉 ≥ f (z∗) − 〈�, z∗〉, (3.8)

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i.e., � is a subgradient of f at z∗ if and only if z∗ is a global minimizer off − 〈�, �〉. For outer � -convex functions, we will replace f (z∗) − 〈�, z∗〉 in theright-hand side of the inequality in (3.8) by f (x) − 〈�, x〉 for some suitablex lying near z∗.

Theorem 3.8. Let X = �n be some n-dimensional normed vector space, D ⊂ Xbe compact and convex, and B ⊂ X be balanced and convex. Let f : D → � beouter � -convex w.r.t. � = B, bounded from below, and lower semicontinuous. Thenfor all z∗ ∈ riD, there is � ∈ �n such that

∃z ∈ (�z∗� + B) ∩ D ∀z ∈ D : f (z) − 〈�, z〉 ≥ f (z) − 〈�, z〉)� (3.9)

In particular, if B = �B(0, ), then there is � ∈ �n such that

∃z ∈ �B(z∗, Js(X )/2) ∩ D ∀z ∈ D : f (z) − 〈�, z〉 ≥ f (z) − 〈�, z〉� (3.10)

Proof. (a) Let X × � = �n+1 be equipped with the product topology.Assume that (xi , �i) ∈ conv(epi f ), i ∈ �, limi→∞ xi = x∗ ∈ �n , andlimi→∞ �i = �∗ ∈ �. We show now that (x∗, �∗) ∈ conv(epi f ). By definitionand Caratheodory’s theorem, (xi , �i) ∈ conv(epi f ) implies the existenceof �ij and (xi

j , �ij) ∈ epi f , j = 1, 2, � � � ,n + 2, such that

�ij ≥ 0,n+2∑j=1

�ij = 1, xi =n+2∑j=1

�ij xij , �i =

n+2∑j=1

�ij�ij

for all i ∈ � and 1 ≤ j ≤ n + 2. Because (�i1, �i2, � � � , �

in+2) and

(xi1, x

i2, � � � , x

in+2) lie in corresponding compact set, we can choose

convergent subsequences, i.e., by using the same notation,

limi→∞

(�i1, �i2, � � � , �

in+2) = (�∗

1, �∗2, � � � , �

∗n+2),

limi→∞

(xi1, x

i2, � � � , x

in+2) = (x∗

1 , x∗2 , � � � , x

∗n+2)�

This yields x∗ = ∑n+2j=1 �∗

j x∗j and

�∗ = limi→∞

�i = limi→∞

n+2∑j=1

�ij�ij ≥

n+2∑j=1

limi→∞

�ij lim infi→∞

�ij =n+2∑j=1

�∗j lim inf

i→∞�ij

because all �ij , i ∈ � and 1 ≤ j ≤ n + 2, are nonnegative. Because (xij , �

ij) ∈

epi f and f is lower semicontinuous and all �∗j , 1 ≤ j ≤ n + 2, are

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nonnegative, it follows

�∗ ≥n+2∑j=1

�∗j lim inf

i→∞f (xi

j ) ≥n+2∑j=1

�∗j f (x

∗j )�

Hence,

(x∗j , f (x∗

j ) + �∗ −n+2∑j=1

�∗j f (x

∗j )

)∈ epi f , 1 ≤ j ≤ n + 2,

and

�∗j ≥ 0,

n+2∑j=1

�∗j = 1, x∗ =

n+2∑j=1

�∗j x

∗j ,

�∗ =n+2∑j=1

�∗j

(f (x∗

j ) + �∗ −n+2∑j=1

�∗j f (x∗

j )

),

i.e., (x∗, �∗) ∈ conv(epi f ). Thus conv(epi f ) is closed.

(b) Let

f (x) := inf�� | (x , �) ∈ conv(epi f )�, x ∈ D�

Because f is bounded from below and conv(epi f ) is closed, we have −∞<f (x) < ∞ for all x ∈ D and epi f = conv(epi f ). By (3.8), for z∗ ∈ riD,there exists � ∈ �n such that

∀z ∈ D : f (z) − 〈�, z〉 ≥ f (z∗) − 〈�, z∗〉� (3.11)

It is trivial if f (z∗) = f (z∗). Therefore, next we have to consider only z∗ ∈riD with f (z∗) > f (z∗), i.e., (z∗, f (z∗)) ∈ conv(epi f )\epi f .

(c) Because f is outer � -convex w.r.t. � = B, by Theorem 3.5, itsepigraph epi f is outer � -convex w.r.t. � = B × �. Following, Theorem 2.4yields that then there exist k ≥ 2 and (xi , �i) ∈ epi f , i = 1, 2, � � � , k, suchthat

(z∗, f (z∗)) ∈ conv�(x1, �1), (x2, �2), � � � , (xk , �k)�

and

(xi − xj , �i − �j) ∈ B × �, 1 ≤ i , j ≤ k�

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It follows that

z∗ =k∑

i=1

�i xi , f (z∗) =k∑

i=1

�i�i for some �i ≥ 0,k∑

i=1

�i = 1, (3.12)

and

xi − xj ∈ B, 1 ≤ i , j ≤ k� (3.13)

By (3.11) and (xi , �i) ∈ epi f , we have

�i − 〈�, xi〉 ≥ f (xi) − 〈�, xi〉 ≥ f (xi) − 〈�, xi〉≥ f (z∗) − 〈�, z∗〉, i = 1, 2, � � � , k, (3.14)

which yields by addition of all k inequalities that

f (z∗) − 〈�, z∗〉 =k∑

i=1

�i(�i − 〈�, xi〉) ≥k∑

i=1

�i(f (xi) − 〈�, xi〉) ≥ f (z∗) − 〈�, z∗〉,

Consequently, all inequalities in (3.14) must be fulfilled as equalities, i.e.,

�i − 〈�, xi〉 = f (xi) − 〈�, xi〉 = f (z∗) − 〈�, z∗〉, i = 1, 2, � � � , k�

These equalities along with (3.11) and f (z) ≥ f (z) imply that

f (z) − 〈�, z〉 ≥ f (z) − 〈�, z〉 ≥ f (z∗) − 〈�, z∗〉 = f (xi) − 〈�, xi〉 (3.15)

holds true for all z ∈ D and i = 1, 2, � � � , k. Because B is convex, (3.12) and(3.13) imply xi ∈ �z∗� + B for all i = 1, 2, � � � , k. Hence, (3.9) is satisfied foreach z ∈ �xi | 1 ≤ i ≤ k�.

(d) If B = �B(0, ), then (3.13) means ‖xi − xj‖ ≤ for 1 ≤ i , j ≤ k.By Proposition 3.1 in Phu [9], there exists z ∈ �xi | 1 ≤ i ≤ k� such that

‖z∗ − z‖ ≤ 12Js(X )�

Hence, (3.15) implies that (3.10) is fulfilled for this z. �

Other formulations for (3.9) and (3.10) are

∀z ∈ D : f (z) − 〈�, z〉 ≥ minz′∈(�z∗�+B)∩D

(f (z ′) − 〈�, z ′〉)

and

∀z ∈ D : f (z) − 〈�, z〉 ≥ minz′∈�B(z∗,Js (X )/2)∩D

(f (z ′) − 〈�, z ′〉),

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respectively. They say that for all z∗ ∈ riD, there exists � ∈ �n such that f +〈�, �〉 admits its minimum on �z∗� + B (or on �B(z∗, Js(X )/2), respectively),which is also global minimum.

If a traditional representation of subgradients like (3.7) is preferred,one can use the equivalent form

∃z ∈ (�z∗� + B) ∩ D ∀z ∈ D : f (z) ≥ f (z) + 〈�, z − z〉)for (3.9) and

∃z ∈ �B(z∗, Js(X )/2) ∩ D ∀z ∈ D : f (z) ≥ f (z) + 〈�, z − z〉for (3.10), i.e., there exists z ∈ (�z∗� + B) ∩ D, or z ∈ �B(z∗, Js(X )/2) ∩ D,respectively, such that the subdifferential �f at x is not empty.

4. CONCLUSIONS

Further properties of outer � -convex sets and outer � -convex functionsare presented in other papers. In particular, the outer -convexityof disturbed functions is studied in Phu [18], where arbitrarily wildbut accordingly bounded perturbations are allowed when some convexfunctions are disturbed.

ACKNOWLEDGMENT

The author would like to express his sincere gratitude to Prof. Dr.Dr. h.c. Eberhard Zeidler for inviting him to the Max Planck Institute forMathematics in the Sciences where parts of this work were done.

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