ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKIY COMPACT CONVEX SET

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ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKLY COMPACT CONVEX SET

This papeer studies the known density theorem of Arrow-Barankin-Blackwell. The following main result is obtained: If X is a Hausdorff locally convex Topological space and C belong to X is a closed convex cone with bounded base, then for every nonempty weakly compact convex subset A, the set of positive proper efficient points of A is dense in the set of efficient points of A.     

Bases of convex cones and Borwein's proper efficiency

In this note, we establish some interesting relationships between the existence of Borwein's proper efficient points and the existence of bases for convex ordering cones in normed linear spaces. We show that, if the closed unit ball in a smooth normed space ordered by a convex cone possesses a proper efficient point in the sense of Borwein, then the ordering cone is based. In particular, a convex ordering cone in a reflexive space is based if the closed unit ball possesses a proper efficient point. Conversely, we show that, in any ordered normed space, if the ordering cone has a base, then every weakly compact set possesses a proper efficient point.The research was conducted while the author was working on his PhD Degree under the supervision of Professor J. M. Borwein, whose guidance and valuable suggestions are gratefully appreciated. The author would like to thank two anonymous referees for their constructive comments and suggestions. This research was supported by an NSERC grant and a Mount Saint Vincent University Research Grant.     

Nonconvex separation theorems and some applications in vector optimization

Separation theorems for an arbitrary set and a not necessarily convex set in a linear topological space are proved and applied to vector optimization. Scalarization results for weakly efficient points and properly efficient points are deduced.     

On the Density of Positive Proper Efficient Points in a Normed Space

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S ⁺, we establish some density results of positive weak* efficient elements of A in E(A, S ⁺).     

The facialQ-topology for compact convex sets

The structure of the set of closed two-sided ideals in aC*-algebraU with identity is described by means of a topology on the set _e K of extreme points of the state spaceK ofU. Recent results of Alfsen, Andersen, Combes, Perdrizet, Wils, and others have shown that such a topology can be defined on the set _e K of extreme points of an arbitrary compact convex subset of a locally convex Hausdorff topological vector space.The structure of the set of closed left ideals in aC*-algebraU with identity can also be described by means of a set of subsets of the set _e K of extreme points of its state spaceK. Akemann, Giles, and Kummer showed that this formed a more general structure than a topology which was called aq-topology. In this paper it is shown that for a reasonably wide class of compact convex subsetsK of locally convex Hausdorff topological vector spaces such aq-topology can also be defined on _e K and that it shares many of the properties of theq-topology defined forC*-algebras. The methods used depend strongly upon recent results of Alfsen and Shultz on the spectral theory of affine functions on compact convex sets.     

Generalization of the Arrow–Barankin–Blackwell Theorem in a Dual Space Setting

In this note, we provide general sufficient conditions under which, if F is a compact [resp. w*-compact] subset of the topological dual Y* of a nonreflexive normed space Y partially ordered by a closed convex pointed cone K, then the set of points in F that can be supported by strictly positive elements in the canonical embedding of Y in Y** is norm dense [resp. w*-dense] in the efficient [maximal] point set of F. This result gives an affirmative answer to the conjecture proposed by Gallagher (Ref. 19), and also generalizes the results stated in Ref. 19 and some space specific results given in Refs. 17, 18, and 11.     

Efficient Sets of Convex Compacta are Arcwise Connected

We prove that the efficient point set Max(Q|K) of a compact convex set QX in a Hausdorff topological vector space X ordered by a closed convex pointed cone KX with nonempty K ⁺ⁱ:={lK\{0}:l(x)>0} is arcwise connected.     

On existence of finite universal Korovkin sets in the centre of group algebras

In this paper, we characterize compact groupsG as well as connected central topological groupsG for which the centreZ(L ¹(G)) admits a finite universal Korovkin set. Also we prove that ifG is a non-connected central topological group which has a compact open normal subgroupK such thatG=KZ, thenZ(L ¹(G)) admits a finite universal Korovkin set if is a finite-dimensional separable metric space or equivalentlyG is separable metrizable andG/K has finite torsion-free rank.     

On the range sets of variational inequalities

This paper investigates the closedness and convexity of the range sets of the variational inequality (VI) problem defined by an affine mappingM and a nonempty closed convex setK. It is proved that the range set is closed ifK is the union of a polyhedron and a compact convex set. Counterexamples are given such that the range set is not closed even ifK is a simple geometrical figure such as a circular cone or a circular cylinder in a three-dimensional space. Several sufficient conditions for closedness and convexity of the range set are presented. Characterization for the convex hull of the range set is established in the case whereK is a cone, while characterization for the closure of the convex hull of the range set is established in general. Finally, some applications to stability of VI problems are derived.This work was supported by the Australian Research Council.We are grateful to Professors M. Seetharama Gowda, Olvi Mangasarian, Jong-Shi Pang, and Steve Robinson for references. We are thankful to Professor Jim Burke for discussions on Theorem 2.1 and Counterexample 3.5.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

On sets having finitely many points of local nonconvexity and propertyP m

If a pointq ofS has the property that each neighborhood ofq contains pointsx andy such that the segmentxy is not contained byS, q is called a point of local nonconvexity ofS. LetQ denote the set of points of local nonconvexity ofS. Tietze’s well known theorem that a closed connected setS in a linear topological space is convex ifQ=φ is generalized in the result:If S is a closed set in a linear topological space such that S ∼ Q is connected and |Q|=n<∞,then S is the union of n+1or fewer closed convex sets. Letk be the minimal number of convex sets needed in a convex covering ofS. Bounds fork in terms ofm andn are obtained for sets having propertyP _m and |Q|=n.     

Super weak compactness and uniform Eberlein compacta

We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull coC of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball B_X* of X^* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space.     

Existence and Density Results for Proper Efficiency in Cone Compact Sets

Existence and density results are established for positive proper efficient points, Henig proper efficient points, and superefficient points in cone compact sets.     

关于Henig真有效性

在局部凸空间中,获得了Henig真有效点的一些等价条件,讨论了Henig真有效点与Benson真有效点之间的关系.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

关于连续映射半群拓扑熵的一点注记

本文研究了度量空间中连续映射构成半群的拓扑熵.利用Patr′ao~([8])的方法,给出了度量空间中两种有限个连续映射构成的半群的拓扑d-熵的定义,比较了两种拓扑d-熵的大小.证明了局部紧致可分度量空间上有限个真映射构成的半群的拓扑d-熵和它的一点紧化空间上对应的拓扑熵相等.上面结果推广了Patr′ao的相应结论.     

On parametric optimization in hilbert spaces

We consider parametric optimization problems of the following type; Minimize p(x,t) subject to x?C, for all t?T, where C is a weakly locally compact, closed, and convex subset of a Hilbert space H, T is a topological space, and p is a real-valued function on H×T lower semicontinuous if H is provided with the weak topology, and continuous in the second variable for all x?H. We investigate how the optimal value, and the set of optimal solutions depend on tET, We apply our results to show that the discrete eigenvalues, and the corresponding normed eigenelements of certain self-adjoint operators in Hilbert spaces depend (the latter strongly) continuous on parameters under reasonable conditions.     

On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE ^w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE ^w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.