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On Dentability and Cones with a Large Dual RACSAM manuscript No. (will be inserted by the editor) On dentability and cones with a large dual Fernando Garc´ıa-Casta~no · M. A. Melguizo Padial Received: date / Accepted: date Abstract In this paper we provide some equivalences on dentability in normed spaces. Among others we prove: the origin is a denting point of a pointed cone C if and only if it is a point of continuity for such a cone and C∗ − C∗ = X∗; x is a denting point of a convex set A if and only if x is a point of continuity and a weakly strongly extreme point of A. We also analize how our results help us to shed some light on several open problems in the literature. Keywords Denting point · point of continuity · dual cone · weakly strongly extreme point Mathematics Subject Classification (2010) MSC 46B40 · MSC 46A40 · MSC 46B20 1 Introduction Throughout the paper X will denote a normed space, k · k the norm of X, X∗ the dual space of X, 0X the origin of X, and R+ the set of nonnegative real numbers. We call weak the weak topology on X and weak ∗ the weak star topology on its dual ∗ X . A nonempty convex subset C of X is called a cone if αC ⊂ C, 8α 2 R+. Fixed a cone C, we define the dual cone for C by C∗ := ff 2 X∗ : f(c) ≥ 0 ; 8c 2 Cg {which is a weak∗-closed subset of X∗{ and the bidual cone for C by C∗∗ := fT 2 The author M. A. Melguizo Padial has been supported by project MTM2017-86182-P (AEI/FEDER, UE). The author Fernando Garc´ıa-Casta~nohas been partially supported by MINECO and FEDER (MTM2014-54182), by Fundaci´onS´eneca - Regi´onde Murcia (19275/PI/14), and by MTM2017-86182-P (AEI/FEDER, UE). Fernando Garc´ıa-Casta~no Departamento de Matem´aticaAplicada. Escuela Polit´ecnicaSuperior. Universidad de Alicante. 03080 San Vicente del Raspeig. Alicante, Spain Tel.: +34 965903400 x Ext. 2823 Orcid: 0000-0002-8352-8235 E-mail: [email protected] M. A. Melguizo Padial Departamento de Matem´aticaAplicada. Escuela Polit´ecnicaSuperior. Universidad de Alicante. 03080 San Vicente del Raspeig. Alicante, Spain 2 Fernando Garc´ıa-Casta~no,M. A. Melguizo Padial X∗∗ : T (f) ≥ 0; 8f 2 C∗g {which is a weak∗-closed subset of X∗∗{. A cone C ⊂ X is said to be quasi-generating (resp. generating) if C − C = X (resp. if C−C = X). It is clear that generating cones are quasi-generating, but the converse is not true (for example, a proper dense subspace). The notions of generating and quasi- generating cones have been widely studied and appear in many results regarding ordered vector spaces, see for example [2,5]. Fixed a normed space X, by qG∗ we will denote the family of cones in X that have a quasi-generating dual cone. Let us introduce the notion of denting point. It connects with the Radon{ Nikod´ymproperty, with renorming theory, and with optimization theory. Before introducing such a notion we need to fix some notation and terminology. A set H is called an open half space of X if it is a weakly open set of the form H = ∗ fx 2 X : f(x) < λg for some f 2 X n f0X∗ g and λ 2 R. We denote H briefly by ff < λg. Let A ⊂ X, a slice of A is a nonempty intersection of A with an open half space of X. We denote by conv(A) (resp. by conv(A)) the convex hull of A (resp. the closed convex hull of A). Besides, we denote by B(x) the open ball with centre x 2 X and radius > 0 and by B(x) the corresponding closed ball. In case x = 0X and = 1 the former set is denoted by BX and the latter by BX . Let A be a convex subset of X, x 2 A is said to be a denting point of A if x 62 conv(AnB(x)), 8 > 0. By the Hahn{Banach theorem, x is a denting point of A if and only if for every > 0 there exists a slice of A containing x with diameter less than . Another notion involved in this work is that of point of continuity. Let A be a convex subset of X, x 2 A is said to be a point of continuity for A if the identity map (A; weak) ! (A; k · k) is continuous at x. The notion of point of continuity is, in general, weaker than the notion of denting point. However, it is still an open problem to clarify if both notions coincide for closed cones in normed spaces. Such a problem raised in [12, Conclusions], paper in which these properties were applied in vector optimization theory (more applications of these properties in vector optimization can be found in [3,8,20]). In [11,14] were given some results characterizing the notion of denting point for a cone in a normed space in terms of the notion of point of continuity plus an extra assumption on the cone. Let us note that such extra assumptions seem not to be related to the closedness of the cone. However, in the context of Banach spaces both notions (dentability and point of continuity) become equivalent for closed cones. The former equivalence was shown in [9] as a direct consequence of a well known characterization of denting points of a closed convex bounded set in a Banach space, namely [18, Theorem]. In this paper we continue the research we began in [11] about the link between dentability and point of continuity in normed spaces (not necessarily complete) and for arbitrary cones (not necessarily closed). In this line of research, we provide now some new results which have a wide range of applicability {as we will see below{, and we solve {among others{ Problem 2.6 we stated in [11]. In the following paragraphs we will present a brief overview of the main results of this paper together with their connections with other results and problems in the literature. Our first contribution is Theorem 2, in which we stablish { among others{ the equivalence between dentability and point of continuity for cones in qG∗. Roughly speaking, the former equivalence can be interpreted as meaning that the notions of denting point and point of continuity in a cone coincide when the dual cone is \large enough". This result generalizes [14, Theorem 4] and, in addition, it provides the following: a negative answer to Gong's question On dentability and cones with a large dual 3 in [12, Conclusions] for cones in qG∗ (see Corollary 1); the equivalence between the following results on density of Arrow, Barankin and Blackwell's type, [12, Theorem 3.2 (a)] and [20, Corollary 4.2], for cones in qG∗ (see Corollary 2); and a positive answer to [14, Problem 5] for cones in qG∗ (see Corollary 3). Luckily, qG∗ is a large class of cones. It contains, for example, the normal cones introduced by Krein in [16]. Normal cones are useful in the theory of ordered topological vector spaces (see [2]) and in the study of extremal problems in differential and integral equations (see [15]). Returning to [11] {paper of which this work is a continuation{, it is worth pointing out that [11, Proposition 2.5] is a cornerstone in the proof of the main theorem in [11]. Such a proposition characterizes the property that the origin is a weakly strongly extreme point of a pointed closed cone. However, it was unknown if the above mentioned characterization remains true if we drop down to the cone the assumption of closedness. Namely, such a question was stated in [11, Problem 2.6]. In this paper we solve the former question in the negative (see Remark 1). Let us note that such a solution is a consequence of Theorem 3 in this paper, which provides a characterization of the property that the origin is a locally weakly strongly extreme point of a pointed (not necessarily closed) cone. As has been said before, the equivalence of dentability and point of continuity for closed cones in Banach spaces was provided by Daniilidis who made use of the characterization [18, Theorem]. Let us point out that such a characterization states that a point of a closed, convex, and bounded set in a Banach space is a denting point if and only if it is a point of continuity and an extreme point for such a set. However, the former equivalence does not work for noncomplete normed spaces (see [19, Example]) and, to our knowledge, a version for normed spaces is required. In this paper, by means of Theorem 4, we provide such a version changing the notion of extreme point by that of weakly strongly extreme point and assuming {only{ the convexity of the involved set. The paper is organized as follows. In Section 2 we state and prove Theorems 2, 3, and 4 mentioned above. In Section 3 we apply the equivalence between dentabil- ity and point of continuity for cones in qG∗ from Theorem 2 to solve {for cones in qG∗{ some open problems in the literature we mentioned before (Gong's question and so on). Finally, in Section 4 we again apply the former equivalence from The- orem 2, but this time to several noncomplete ordered normed spaces. As a result, we obtain some geometric properties of their corresponding canonical order cones. 2 Results on dentability The main objective in this section is to state and prove Theorems 2, 3, and 4 just commented in the introduction.
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