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Extremality of Convex Sets with Some Applications

In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new calculus results on intersection rules for normal cones to convex sets and on infimal convolutions of support functions.


Introduction
Convex analysis has been well recognized as an important area of mathematics with numerous applications to optimization, control, economics, and many other disciplines. We refer the reader to the fundamental monographs [3,4,11,16,21] and the bibliographies therein for various aspects of convex analysis and its applications. Jon Borwein, who unexpectedly passed away on August 2, 2016, made pivotal contributions to these and related fields of Applied Mathematics, among other areas of his fantastic creative activity.
Methods and constructions of convex analysis play also a decisive role in the study of nonconvex functions and sets by using certain convexification procedures. In particular, calculus and applications of Clarke's generalized gradients for nonconvex functions [9] is based on appropriate convexifications and employing techniques and results of convex analysis.
Besides this, other ideas have been developed in the study and applications of nonconvex functions, sets, and set-valued mappings in the framework of variational analysis, which employs variational/optimization principles married to perturbation and approximation techniques; see the books [7,12,18] for extended expositions in finite and infinite dimensions. Powerful tools, results, and applications of variational analysis have been obtained by using the dual-space geometric approach [12] based on the extremal principle (a geometric variational principle) for systems of sets. This approach produces first a full calculus of generalized normals to nonconvex sets and then applies it to establish comprehensive calculus rules for related subgradients of extended-real-valued functions and coderivatives of setvalued mappings. Needless to say that well-developed calculus of generalized differentiation is an unavoidable requirement and the key for various applications.
Addressing generally nonconvex objects, results of variational analysis contain corresponding convex facts as their particular cases. However, basic variational techniques involving limiting procedures do not fully capture advantages from the presence of convexity. Indeed, the major calculus results of [12] hold in Asplund spaces (i.e., such Banach spaces where every separable subspace has a separable dual) and the closedness of sets (epigraphs for extended-real-valued function, graphs for set-valued mappings) is a standing assumption.
The major goal of this paper is to investigate a counterpart of the variational geometric approach to the study of convex sets in locally convex topological vector (LCTV) spaces without any completeness and closedness assumptions. Based on an enhanced notion of set extremality, which is a global version of the corresponding local concept largely developed and applied in [12] while occurring to be particularly useful in the convex setting mainly exploited here, this approach allows us to obtain the basic intersection rule for normals to convex sets under a new qualification condition. The same approach also allows us to derive new calculus results for support functions of convex set intersections in general LCTV spaces. Note that these results can be used to obtain major calculus rules of generalized differentiation and Fenchel conjugates for extended-real-valued convex functions; cf. our previous publications [13,14] for some versions in finite dimensions.
The rest of the paper is organized as follows. In Section 2 we introduce the aforementioned version of set extremality, establish its relationships with the separation property for convex sets, and derive various extremality conditions. The obtained results are applied in Section 3 to get the normal cone representation for convex set intersections under a new qualification condition. In Section 4 this approach is employed to represent the support function of set intersections via the infimal convolution of supports to intersection components.
For simplicity of presentation we suppose, unless otherwise stated, that all the spaces under consideration are normed linear spaces. The reader can check that the results obtained below in this setting hold true in the LCTV space generality.
The notation used throughout the paper is standard in the areas of functional, convex, and variational analysis; cf. [12,16,18,21]. Recall that the closed ball centered atx with radius r > 0 is denoted by B(x; r) while the closed unit ball of the space X in question and its topological dual X * are denoted by B and B * , respectively, if no confusion arises. Given a convex set Ω ⊂ X, we write R + (Ω) := {tv ∈ X| t ∈ R + , v ∈ Ω}, where R + signifies the collection of positive numbers, and use the symbol Ω for the topological closure of Ω. Finally, remind the notation for the (algebraic) core of a set: core Ω := x ∈ Ω ∀ v ∈ X ∃ γ > 0 such that x + tv ∈ Ω whenever |t| < γ .

Extremal Systems of Sets
We start this section with the definition of extremality for set systems, which is inspired by the notion of local set extremality in variational analysis (see [12, Definition 2.1]) while having some special features that are beneficial for convex sets. In particular, we do not require that the sets have a common point.
Definition 2.1 (set extremality). We say that two nonempty sets Ω 1 , Ω 2 ⊂ X form an extremal system if for any ε > 0 there exists a ∈ X such that a ≤ ε and (Ω 1 + a) ∩ Ω 2 = ∅. (2.1) Observe similarly to [12] that the notion of set extremality introduced in Definition 2.1 covers (global) optimal solutions to problems of constrained optimization with scalar, vector, and set-valued objectives, various equilibrium concepts arising in operations research, mechanics, and economic modeling, etc. Furthermore, the set extremality naturally arises in deriving calculus rules of generalized differentiation in variational analysis. In particular, we are going to demonstrate this below in our device of the normal cone intersection rule and the support function representation for convex set intersections presented in the paper.
To prove the final assertion (iii), take x * = 0 from (2.3) and find c ∈ X such that x * , c > 0. For any ε > 0 we can select a := −c/k satisfying a < ε when k ∈ IN is sufficiently large. Let as show that (2.1) holds with this vector a. If it is not the case, then there exists x ∈ Ω 2 such that x − a ∈ Ω 1 . By the separation property (2.3) we have which gives us by the above construction of a ∈ X that and therefore x * , c ≤ 0. It contradicts the choice of c ∈ X and hence justifies assertion (iii) while completing in this way the proof of the theorem.
Proof. It is well known that the assumptions imposed in the corollary ensure the separation property for convex sets. Thus the set extremality of Ω 1 , Ω 2 follows from Theorem 2.2(iii).
To verify the last assertion of the corollary, take anyx ∈ int Ω 1 and find r > 0 such that int B(x; r) ⊂ Ω 1 . Then for any fixed point x ∈ Ω 2 we have and thus int( As established in Theorem 2.2(ii), the set extremality in (2.1) implies the separation property (2.3) and its equivalent form (2.4) wheneverx ∈ Ω 1 ∩ Ω 2 under the nonempty difference interior int(Ω 1 − Ω 2 ) = ∅ for arbitrary convex sets Ω 1 , Ω 2 in LCTV spaces. Could we relax this assumption? The next theorem shows that it can be done, for closed convex subsets of Banach spaces, in both approximate and exact forms of the convex extremal principle. Furthermore, the results obtained therein justify that both of these forms are characterizations of the convex set extremality under the SNC property of one of the sets involved without imposing any interiority assumption on them or their difference.
To proceed, recall first the definition of the SNC property used below for convex sets; compare it with a nonconvex counterpart from [12,Definition 1.20] where the normal cone is taken from (2.2), and where the symbol w * → signifies the sequential convergence in the weak * topology of X * . We have already mentioned in Remark 2.4 the explicit description of the SNC property for closed convex sets with nonempty relative interiors in Banach spaces given in [12,Theorem 1.21]. Assertion (ii) of the next theorem employs SNC (2.5) for furnishing the limiting procedure in general Banach spaces.
Theorem 2.5 (approximate and exact versions of the convex extremal principle in Banach spaces). Let Ω 1 and Ω 2 be closed convex subsets of a Banach space X, and let x be any common point of Ω 1 , Ω 2 . Consider the following assertions: (i) The sets Ω i , i = 1, 2, form an extremal system in X.
Define the convex, lower semicontinuous, and bounded from below function f : via the indicator function of the closed set Ω 1 × Ω 2 . It follows from (2.1) that f (x 1 , x 2 ) > 0 on X × X and f (x,x) = a ≤ ε 2 for anyx ∈ Ω 1 × Ω 2 . Applying to (2.7) the Ekeland variational principle (see, e.g., [12, Theorem 2.26(i)]), we find a pair ( The latter means that the function ϕ(x 1 , Thus the generalized Fermat rule tells us that 0 ∈ ∂ϕ(x 1ε , x 2ε ). Taking into account the summation structure of f in (2.7), we apply to its subdifferential the classical Moreau-Rockafellar theorem that allows us to find-by standard subdifferentiation of the norm and indicator functions-such dual elements x * iε ∈ N (x iε ; Ω i ) + εB * for i = 1, 2 that all the conditions in (2.6) are satisfied. This justifies assertion (ii) of the theorem.
We verify next the validity of (ii)=⇒(iii) by furnishing the passage to the limit in (2.6) as ε ↓ 0 with the help of the SNC property of, say, the set Ω 1 atx. Take a sequence ε k ↓ 0 as k → ∞ and find by (2.6) the corresponding septuples (x 1k , x 2k , x * k , x * 1k , x * 2k , e * 1k , e * 2k ) so that x 1k →x, x 2k →x as k → ∞, and for all k ∈ IN and i = 1, 2. The classical Banach-Alaoglu theorem of functional analysis tells us that for any Banach space X the sequence of triples (x * k , e * 1k , e * 2k ) contains a subnet converging to some (x * , e * 1 , e * 2 ) ∈ B * × B * × B * in the weak * topology of X * . It follows from (2.8) and definition (2.2) of the normal cone to convex sets that the corresponding subnets of {x * 1k , x * 2k )} converge in the latter topology to some pair (x * 1 , x * 2 ) ∈ X * × X * satisfying x * 1 = −x * 2 = x * and x * i ∈ N (x; Ω i ) for i = 1, 2. To justify (iii), it remains to show that we can always find x * = 0 in this way provided that Ω 1 is SNC atx. Assuming the contrary, let us first check that {x * 1k } converges to zero in the weak * topology. If it is not the case, there is z ∈ X such that the numerical sequence { x * 1k , z } does not converge to zero. Fix w ∈ Ω 1 and for each k ∈ IN consider the set which is a neighborhood of x * 1 in the weak * topology of X * . By extracting numerical subsequences in (2.9), suppose without loss of generality that Remembering that x * 1k ∈ N (x 1k ; Ω 1 ) by (2.8) gives us the estimate (2.8). Passing now to the limit in (2.10) tells us that x * 1 , w −x ≤ 0 and so x * 1 ∈ N (x; Ω 1 ). It follows from (2.8) that −x * 1 ∈ N (x; Ω 2 ) and thus x * 1 ∈ N (x; Ω 1 ) ∩ (−N (x; Ω 2 )) = {0}, which contradicts the imposed assumption on x * 1k , z → 0. Therefore the sequence {x * 1k } converges to zero in the weak * topology of X * , which implies its sequential convergence x * To check finally the equivalence assertion in (iii), observe that the separation property (2.3) ensures by Theorem 2.2(iii) that the sets Ω 1 , Ω 2 form an extremal system in X, and so we have conditions (2.6) in (i). Since implication (i)=⇒(ii) has been verified above, this readily justifies the claimed equivalences in (iii) and thus completes the proof of theorem.
As an immediate consequence of the (convex) approximate extremal principle in Theorem 2.5(ii), we obtain the celebrated Bishop-Phelps theorem for closed convex sets in general Banach spaces; see [15,Theorem 3.18]. Recall thatx ∈ Ω is a support point of Ω ⊂ X if there is 0 = x * ∈ X * such that the function x → x * , x attaints its supremum on Ω atx.

Corollary 2.6 (Bishop-Phelps theorem).
Let Ω be a nonempty, closed, and convex subset of a Banach space X. Then the support points of Ω are dense on the boundary of Ω.
Proof. It is obvious from (2.1) and the definition of boundary points that for any boundary pointx of Ω, the sets Ω 1 := {x} and Ω 2 := Ω form an extremal system in X. Then the result follows from (2.6) and the normal cone structure in (2.2).
Note that a geometric approach involving the approximate extremality conditions (2.6) at points nearby may be useful for applications to the so-called sequential convex subdifferential calculus initiated by Attouch-Baillon-Théra [1] and Thibault [20] in different frameworks and then developed in other publications. Likewise it can be applied as a geometric device of coderivative and conjugate calculus rules, which is our intention in the future research.

Normal Cone Intersection Rule
In this section we employ the set extremality and the results of Theorem 2.2 to obtain the exact intersection rule for the normal cone (2.2) under a new qualification condition.
The following theorem justifies a precise representation of the normal cone N (x; Ω 1 ∩Ω 2 ) via normals to each sets Ω 1 and Ω 2 under the new qualification condition (3.1) depending onx, which is weaker than the standard interiority condition in LCTV spaces. For convenience we refer to (3.1) as to the bounded extremality condition.
Dividing both sides of the obtained inequality by γ < 0, we arrive at Remark 3.2 (comparing qualification conditions for the normal intersection formula). We have the following useful observations: (i) It is easy to see that, if one of the sets Ω 1 , Ω 2 is bounded, the introduced qualification condition (3.1) reduces to the difference interiority condition Furthermore, (3.1) surely holds under the validity of the classical interiority condition Ω 1 ∩ (int Ω 2 ) = ∅, which is the only condition previously known to us that ensures the validity of the intersection formula (3.2) in the general LCTV (or even normed) space setting. Indeed, if the latter condition is satisfied, take u ∈ Ω 1 ∩ (int Ω 2 ) and γ > 0 such that u + γB ⊂ Ω 2 . Then we choose r > 0 with u + γB ⊂ Ω 2 ∩ B(x; r). Thus γB ⊂ Ω 1 − (Ω 2 ∩ B(x; r)) and so 0 ∈ int(Ω 1 − (Ω 2 ∩ V )), where V := B(x; r).
(ii) If X is Banach and both sets Ω 1 , Ω 2 are closed with int(Ω 1 − Ω 2 ) = ∅, the difference interiority condition (3.6) reduces to Rockafellar's core qualification condition 0 ∈ core(Ω 1 − Ω 2 ) introduced in [17]. This follows from the equivalence valid in this case. Indeed, the implication "⇐=" in (3.7) is obvious due to int Ω ⊂ core Ω for any set. To verify the opposite implication in (3.7), recall that int Ω = core Ω for closed convex subsets of Banach spaces by [7,Theorem 4.1.8]. Using now the well-known fact that int Ω = int Ω for convex sets with nonempty interiors yields Note that the core qualification condition is superseded in the same setting by the requirement that R + (Ω 1 − Ω 2 ) ⊂ X is a closed subspace, which is known as the Attouch-Brézis regularity condition established in [2] with the usage of convex duality and the fundamental Banach-Dieudonné-Krein-Šmulian theorem in general Banach spaces.
For any x ∈ X denote u := γ x +1 x ∈ γB and get u = w 1 − w 2 with w i ∈ Ω i for i = 1, 2. Hence there is a constantγ > 0 depending on x and r for which This readily justifies the relationships for all 0 < t <γ and thus establishes the claimed inclusion (3.8) by the core definition (1.1).

Support Functions for Set Intersections
In this section we derive a precise representation of support functions for convex set intersections via the infimal convolution of the support functions to the intersection components under the difference interiority condition (3.6). This result under (3.6) seems to be new in the literature on convex analysis in LCTV (and also in normed) spaces; see Remark 4.2 for more discussions. Furthermore, we present a novel geometric device for results of this type by employing set extremality and the normal intersection rule obtained above.
Recall that the support function of a nonempty set Ω ⊂ X is given by (4.1) The infimal convolution of two functions f, g : X → R is Theorem 4.1 (support functions for set intersections via infimal convolutions).
Let the sets Ω 1 , Ω 2 ⊂ X be nonempty and convex, and let and one of them be bounded. Then the difference interiority condition (3.6) ensures the representation Moreover, for any x * ∈ dom (σ Ω 1 ∩Ω 2 ) there are x * 1 , x * 2 ∈ X * such that x * = x * 1 + x * 2 and Proof. First we check that the inequality "≤" in (4.3) holds in the general setting. Fix any x * ∈ X * and pick x * 1 , x * 2 ∈ X * such that x * = x * 1 + x * 2 . Then it follows from (4.1) that Taking the infimum on the right-hand side above with respect to all x * 1 , x * 2 ∈ X * satisfying x * 1 + x * 2 = x * gives us by definition (4.2) of the infimal convolution that This verifies and the inequality "≤" in (4.3) by taking the supremum on the left-hand side therein with respect to x ∈ Ω 1 ∩ Ω 2 .
To justify further the opposite inequality in (4.3) under the validity of (3.6), suppose that Ω 2 is bounded. It suffices to consider the case where x * ∈ dom (σ Ω 1 ∩Ω 2 ) and prove the inequality "≤" in (4.4); then the one in (4.3) and both statements of the theorem follow.
Denoting x * 1 := y * /β + x * and x * 2 := −y * /β, we have x * 1 + x * 2 = x * and x * 1 , x + x * 2 , y ≤ α for all x ∈ Ω 1 and y ∈ Ω 2 . This shows that σ Ω 1 (x * 1 ) + σ Ω 2 (x * 2 ) ≤ α = σ Ω 1 ∩Ω 2 (x * ) and thus completes the proof of the theorem. x ∈ X , x * ∈ X * , of the indicator function f (x) := δ(x; Ω) of a given set Ω ⊂ X, and hence a well-developed conjugate calculus can be applied to establish representations (4.3) and (4.4); see, e.g., the books [8,17,19,21] and the references therein. However, it seems to us that such an approach from Fenchel duality misses the specific results of Theorem 4.1 derived for the support function under the qualification condition (3.6) in general LCTV spaces. Observe also that, in contrast to analytical schemes usually applied to deriving conjugate calculus and then deducing results of the type of Theorems 3.1 and 4.1 from them, we develop here a geometric approach in the other direction based on set extremality.