scholarly journals On a theorem of Dvoretsky, Wald, and Wolfowitz concerning Liapounov Measures

1987 ◽  
Vol 29 (2) ◽  
pp. 205-220 ◽  
Author(s):  
D. A. Edwards
Keyword(s):  
Natural Number ◽  
Convex Subset ◽  
Vector Measure ◽  
Compact Convex ◽  
Partition Of Unity ◽  
Compact Convex Subset ◽  
Vector Spaces ◽  
Measurable Partition ◽  
Infinite Dimensional ◽  
Image Position

Let ω be a non-empty set, ℱ a Boolean σ-algebra of subsets of Ω, k a natural number, and let m:ℱ→ℝk be a non-atomic vector measure. Then, by the celebrated theorem of Liapounov [11], the range m[3F] = {m(A): A ε ℱ3F} of m is a compact convex subset of ℝk. This theorem has been generalized in a number of ways. For example Kingman and Robertson [8] and Knowles [9] have shown that, under appropriate conditions, results in the same spirit can be proved for measures taking their values in infinite-dimensional vector spaces. Another type of generalization was obtained by Dvoretsky, Wald and Wolfowitz [6,7]. What they do is to take m as above together with a natural number n≥ 1. They then consider the set Knof all vectorswhere (A1 A2,…, An) is an ordered ℱ-measurable partition of Ω (i.e. a partition whose terms A, all belong to ℱ). They prove in [6] that Kn is a compact convex subset of ℝnk and moreover that Kn is equal to the set of all vectors of the formwhere (ϕ1, ϕ2…, ϕn) is an ℱ-measurable partition of unity; i.e. it is an n-tuple of non-negative ϕr on Ω such thatLiapounov's theorem can be obtained as a corollary of this result by taking n= 2.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

scholarly journals Some remarks on the location of fixed points

1984 ◽  
Vol 37 (3) ◽  
pp. 358-365
Author(s):  
Michael Edelstein ◽  
Daryl Tingley
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Convex Subset ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Chebyshev Center ◽  
Weakly Compact ◽  
Asymptotic Center

AbstractSeveral procedures for locating fixed points of nonexpansive selfmaps of a weakly compact convex subset of a Banach space are presented. Some of the results involve the notion of an asymptotic center or a Chebyshev center.

Some Inequalities Related to Planar Convex Sets

1982 ◽  
Vol 25 (3) ◽  
pp. 302-310 ◽  
Author(s):  
R. J. Gardner ◽  
S. Kwapien ◽  
D. P. Laurie
Keyword(s):  
Convex Subset ◽  
Convex Sets ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Planar Convex

AbstractB. Grünbaum and J. N. Lillington have considered inequalities defined by three lines meeting in a compact convex subset of the plane. We prove a conjecture of Lillington and propose some conjectures of our own.

scholarly journals On Bare Points

1969 ◽  
Vol 9 (1-2) ◽  
pp. 25-28 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Normed Linear Space ◽  
Compact Convex ◽  
Proper Subset ◽  
Compact Convex Subset

This note shows that the set of bare points of a compact convex subset of a normed linear space is, in general, a proper subset of its set of exposed points.

scholarly journals Infinite dimensional representations of

1990 ◽  
Vol 32 (1) ◽  
pp. 25-33 ◽  
Author(s):  
A. Dean ◽  
F. Zorzitto
Keyword(s):  
Dynkin Diagram ◽  
Finite Dimension ◽  
Vector Spaces ◽  
Closed Field ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Image Position ◽  
Subspace Problem ◽  
Extended Dynkin Diagram

By a representation of the extended Dynkin diagram we shall mean a list of 5 vector spaces P, E1, E2, E3, E4 over an algebraically closed field K, and 4 linear maps a1, a2, a3, a4 as shown.The spaces need not be of finite dimension.In their solution of the 4-subspace problem [6], Gelfand and Ponomarev have classified such representations when the spaces are finite dimensional. A representation like (1) can also be viewed as a module over the K-algebra R4 consisting of all 5 × 5 matrices having zeros off the first row and off the main diagonal.

scholarly journals Revisiting Cauty's proof of the Schauder conjecture

2003 ◽  
Vol 2003 (7) ◽  
pp. 407-433 ◽  
Author(s):  
Tadeusz Dobrowolski
Keyword(s):  
Fixed Point ◽  
Detailed Analysis ◽  
Linear Space ◽  
Convex Subset ◽  
Approximation Property ◽  
Compact Convex ◽  
Fixed Point Property ◽  
Compact Convex Subset ◽  
Point Property ◽  
Linear Spaces

The Schauder conjecture that every compact convex subset of a metric linear space has the fixed-point property was recently established by Cauty (2001). This paper elaborates on Cauty's proof in order to make it more detailed, and therefore more accessible. Such a detailed analysis allows us to show that the convex compacta in metric linear spaces possess the simplicial approximation property introduced by Kalton, Peck, and Roberts. The latter demonstrates that the original Schauder approach to solve the conjecture is in some sense “correctable.”

scholarly journals Generalizations of F. E. Browder's sharpened form of the schauder fixed point theorem

1987 ◽  
Vol 42 (3) ◽  
pp. 390-398 ◽  
Author(s):  
Kok-Keong Tan
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Subset ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Hausdorff Topological Vector Space ◽  
Coincidence Theorem ◽  
Nonempty Compact ◽  
Nonempty Compact Convex Subset ◽  
Locally Convex

AbstractLet E be a Hausdorff topological vector space, let K be a nonempty compact convex subset of E and let f, g: K → 2E be upper semicontinuous such that for each x ∈ K, f(x) and g(x) are nonempty compact convex. Let Ω ⊂ 2E be convex and contain all sets of the form x − f(x), y − x + g(x) − f(x), for x, y ∈ K. Suppose p: K × Ω →, R satisfies: (i) for each (x, A) ∈ K × Ω and for ε > 0, there exist a neighborhood U of x in K and an open subset set G in E with A ⊂ G such that for all (y, B) ∈ K ×Ω with y ∈ U and B ⊂ G, | p(y, B) - p(x, A)| < ε, and (ii) for each fixed X ∈ K, p(x, ·) is a convex function on Ω. If p(x, x − f(x)) ≤ p(x, g(x) − f(x)) for all x ∈ K, and if, for each x ∈ K with f(x) ∩ g(x) = ø, there exists y ∈ K with p(x, y − x + g(x) − f(x)) < p(x, x − f(x)), then there exists an x0 ∈ K such that f(x0) ∩ g(x0) ≠ ø. Another coincidence theorem on a nonempty compact convex subset of a Hausdorff locally convex topological vector space is also given.

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