Mazur Intersection Properties for Compact and Weakly Compact Convex Sets

1998 ◽  
Vol 41 (2) ◽  
pp. 225-230 ◽  
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.

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

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

2013 ◽  
Vol 56 (2) ◽  
pp. 272-282 ◽  
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.

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

1982 ◽  
Vol 25 (3) ◽  
pp. 339-343 ◽  
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.

Rates of convergence for random approximations of convex sets

1996 ◽  
Vol 28 (02) ◽  
pp. 384-393 ◽  
Author(s):  
Lutz Dümbgen ◽  
Günther Walther
Keyword(s):  
Convex Subset ◽  
Hausdorff Distance ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Almost Sure Convergence ◽  
Rates Of Convergence ◽  
Random Sets ◽  
Compact Convex Set

The Hausdorff distance between a compact convex set K ⊂ ℝd and random sets is studied. Basic inequalities are derived for the case of being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of being the intersection of a random family of halfspaces containing K.

Exact distributions for shapes of random triangles in convex sets

1985 ◽  
Vol 17 (02) ◽  
pp. 308-329 ◽  
Author(s):  
D. G. Kendall
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Direct Proof ◽  
Circular Disk ◽  
Compact Convex Set ◽  
Exact Evaluation ◽  
Exact Distributions ◽  
New Formula ◽  
Relevant Section

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

Exact distributions for shapes of random triangles in convex sets

1985 ◽  
Vol 17 (2) ◽  
pp. 308-329 ◽  
Author(s):  
D. G. Kendall
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Direct Proof ◽  
Circular Disk ◽  
Compact Convex Set ◽  
Exact Evaluation ◽  
Exact Distributions ◽  
New Formula ◽  
Relevant Section

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

A geometric characterization concerning compact, convex sets

1991 ◽  
Vol 109 (2) ◽  
pp. 351-361 ◽  
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.

Weakly prime compact convex sets and uniform algebras

1977 ◽  
Vol 81 (2) ◽  
pp. 225-232 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Uniform Algebra ◽  
Reduction Theory ◽  
Natural Analogue ◽  
Uniform Algebras ◽  
Compact Convex Set ◽  
Analytic Sets

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.

scholarly journals Additive selections and the stability of the Cauchy functional equation

2003 ◽  
Vol 44 (3) ◽  
pp. 323-337 ◽  
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.

scholarly journals Circumradius-diameter and width-inradius relations for lattice constrained convex sets

1999 ◽  
Vol 59 (1) ◽  
pp. 147-152 ◽  
Author(s):  
Poh Wah Awyong ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Integer Lattice ◽  
Lattice Points ◽  
Compact Convex Set ◽  
Planar Convex ◽  
Rectangular Lattices

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

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