A contractive fixed point free mapping on a weakly compact convex set

2014 ◽  
Vol 223 (3) ◽  
pp. 275-283 ◽  
Author(s):  
Jared Burns ◽  
Chris Lennard ◽  
Jeromy Sivek
Keyword(s):  
Fixed Point ◽  
Convex Set ◽  
Compact Convex ◽  
Weakly Compact ◽  
Compact Convex Set

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.

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.

The Embedding of Compact Convex Sets in Locally Convex Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 449-454 ◽  
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.

scholarly journals The range of an o-weakly compact mapping

1985 ◽  
Vol 39 (3) ◽  
pp. 391-399 ◽  
Author(s):  
P. G. Dodds
Keyword(s):  
Convex Set ◽  
Vector Measure ◽  
Compact Convex ◽  
Locally Convex Space ◽  
Weakly Compact ◽  
Compact Mapping ◽  
Compact Convex Set ◽  
Locally Convex ◽  
Order Continuous

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.

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 Extreme point properties of fixed-point sets

1972 ◽  
Vol 6 (2) ◽  
pp. 241-249 ◽  
Author(s):  
Rodney Nillsen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Extreme Point ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Topological Properties ◽  
Compact Convex Set ◽  
Continuous Maps ◽  
Fixed Point Sets

We consider a semigroup S acting as affine continuous maps on a compact convex set X. F denotes the corresponding set of fixed points. Let exX and exF denote the corresponding sets of extreme points. If X is a simplex, conditions are given which ensure that when x ε F, the maximal measure representing x invariant under S. We also prove exF = F ∩ exX under conditions involving extreme amenability of S. Topological properties of exF are also studied.

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (02) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random setsX, densitiesDφ(X) of additive functionalsφare defined and formulas forare derived whenKis a compact convex set in. In particular, for the quermassintegrals and motioninvariantX, these formulas are in analogy with classical integral geometric formulas. The case whereXis the union set of a Poisson processYof convex particles is considered separately. Here, formulas involving the intensity measure ofYare obtained.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (2) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.

Hyperplans poissoniens et compacts de Steiner

1974 ◽  
Vol 6 (03) ◽  
pp. 563-579 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Stationary Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Isotropic Case ◽  
Geometrical Properties ◽  
Steiner Formula ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Linear Variety

A compact convex set in RN is Steiner if it is a finite Minkowski sum of line segments, or a limit of such finite sums, and then satisfies an extension of the Steiner formula. With each Poisson hyperplane stationary process A is uniquely associated a Steiner set M, and for any linear variety V, the Steiner set associated with is the projection of M on V. The density of the order k network Ak (i.e., the set of the intersections of k hyperplanes belonging to A) is linked with simple geometrical properties of M. In the isotropic case, the expression of the covariance measures associated with Ak is derived and compared with the analogous results obtained for (N — k)-dimensional Poisson flats.

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