The range of an o-weakly compact mapping

AbstractIt is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

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A geometric characterization concerning compact, convex sets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100069814 ◽

1991 ◽

Vol 109 (2) ◽

pp. 351-361 ◽

Cited By ~ 2

Author(s):

Christopher J. Mulvey ◽

Joan Wick Pelletier

Keyword(s):

Normed Space ◽

Convex Sets ◽

Convex Set ◽

Compact Convex ◽

Necessary And Sufficient Condition ◽

Categorical Logic ◽

Compact Convex Set ◽

Necessary And Sufficient ◽

Formal Statement

In this paper, we are concerned with establishing a characterization of any compact, convex set K in a normed space A in an arbitrary topos with natural number object. The characterization is geometric, not in the sense of categorical logic, but in the intuitive one, of describing any compact, convex set K in terms of simpler sets in the normed space A. It is a characterization of the compact, convex set in the sense that it provides a necessary and sufficient condition for an element of the normed space to lie within it. Having said this, we should immediately qualify our statement by stressing that this is the intuitive content of what is proved; the formal statement of the characterization is required to be in terms appropriate to the constructive context of the techniques used.

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Additive selections and the stability of the Cauchy functional equation

The ANZIAM Journal ◽

10.1017/s1446181100008051 ◽

2003 ◽

Vol 44 (3) ◽

pp. 323-337 ◽

Cited By ~ 11

Author(s):

Roman Badora ◽

Roman Ger ◽

Zsolt Páles

Keyword(s):

Convex Set ◽

Compact Convex ◽

Necessary And Sufficient Condition ◽

Cauchy Functional Equation ◽

Amenable Semigroup ◽

Weakly Compact ◽

Compact Convex Set ◽

The Stability ◽

Necessary And Sufficient ◽

Selection Of

AbstractThe main result of this paper offers a necessary and sufficient condition for the existence of an additive selection of a weakly compact convex set-valued map defined on an amenable semigroup. As an application, we obtain characterisations of the solutions of several functional inequalities, including that of quasi-additive functions.

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On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-169-3 ◽

2013 ◽

Vol 56 (2) ◽

pp. 272-282 ◽

Cited By ~ 4

Author(s):

Lixin Cheng ◽

Zhenghua Luo ◽

Yu Zhou

Keyword(s):

Banach Space ◽

Convex Sets ◽

Convex Set ◽

Compact Convex ◽

Lower Semicontinuous ◽

Weakly Compact ◽

Bounded Set ◽

Fréchet Differentiable ◽

Bounded Convex

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.

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A Note on Asymptotic Normal Structure and Close-to-Normal Structure

Canadian Mathematical Bulletin ◽

10.4153/cmb-1982-047-3 ◽

1982 ◽

Vol 25 (3) ◽

pp. 339-343 ◽

Cited By ~ 2

Author(s):

Kok-Keong Tan

Keyword(s):

Banach Space ◽

Convex Subset ◽

Convex Set ◽

Reflexive Banach Space ◽

Compact Convex ◽

Normal Structure ◽

Weakly Compact ◽

Compact Convex Set ◽

Asymptotic Normal ◽

Open Question

AbstractA closed convex subset X of a Banach space E is said to have (i) asymptotic normal structure if for each bounded closed convex subset C of X containing more than one point and for each sequence in C satisfying ‖xn − xn + 1‖ → 0 as n → ∞, there is a point x ∈ C such that ; (ii) close-to-normal structure if for each bounded closed convex subset C of X containing more than one point, there is a point x ∈ C such that ‖x − y‖ < diam‖ ‖(C) for all y ∈ C While asymptotic normal structure and close-to-normal structure are both implied by normal structure, they are not related. The example that a reflexive Banach space which has asymptotic normal structure but not close-to normal structure provides us a non-empty weakly compact convex set which does not have close-to-normal structure. This answers an open question posed by Wong in [9] and hence also provides us a Kannan map defined on a weakly compact convex set which does not have a fixed point.

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The Embedding of Compact Convex Sets in Locally Convex Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-038-0 ◽

1978 ◽

Vol 30 (03) ◽

pp. 449-454 ◽

Cited By ~ 5

Author(s):

James W. Roberts

Keyword(s):

Fixed Point ◽

Fixed Point Theorems ◽

Convex Sets ◽

Convex Set ◽

Compact Convex ◽

Locally Convex Spaces ◽

Hausdorff Topological Vector Space ◽

Compact Convex Set ◽

Affine Functions ◽

Locally Convex

In studying compact convex sets it is usually assumed that the compact convex set X is contained in a Hausdorff topological vector space L where the topology on X is the relative topology. Usually one assumes that L is locally convex. The reason for this is that most of the major theorems such as the Krein-Milman, Choquet-Bishop-de Leeuw, and most of the fixed point theorems require that there be enough continuous affine functions on X to separate points.

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Mazur Intersection Properties for Compact and Weakly Compact Convex Sets

Canadian Mathematical Bulletin ◽

10.4153/cmb-1998-032-8 ◽

1998 ◽

Vol 41 (2) ◽

pp. 225-230 ◽

Cited By ~ 2

Author(s):

Jon Vanderwerff

Keyword(s):

Banach Space ◽

Convex Sets ◽

Convex Set ◽

Compact Convex ◽

Weakly Compact ◽

Compact Convex Set

AbstractVarious authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively.

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