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 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.”

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.

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.

scholarly journals An application of KKM-map principle

1992 ◽  
Vol 15 (4) ◽  
pp. 659-661 ◽  
Author(s):  
A. Carbone
Keyword(s):  
Convex Subset ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Compact Convex ◽  
Nonempty Convex Subset ◽  
Nonempty Convex ◽  
Coincidence Theorems ◽  
Nonempty Compact ◽  
Nonempty Compact Convex Subset ◽  
Kkm Map

The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. LetCbe a nonempty convex subset of a normed linear spaceX,f:C→Xa continuous function,g:C→Ccontinuous, onto and almost quasi-convex. Assume thatChas a nonempty compact convex subsetDsuch that the setA={y∈C:‖g(x)−f(y)‖≥‖g(y)−f(y)‖   for   all   x∈D}is compact.Then there is a pointy0∈Csuch that‖g(y0)−f(y0)‖=d(f(y0),C).

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.

On a Question of Colin Clark Concerning Three Properties of Convex Sets(1)

1972 ◽  
Vol 15 (4) ◽  
pp. 535-537
Author(s):  
Victor Klee
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Upper Bound ◽  
Normed Linear Space ◽  
Convex Sets ◽  
Finite Width

Let C be a convex subset of a normed linear space E. The following properties of C were studied by Clark [2]:(1)C is of finite width/ that is, C lies between two parallel closed hyperplanes;(2)C is of finite width in some direction; that is, for some line L in E there is a finite upper bound for the lengths of C's intersections with lines parallel to L;(3)C is partially bounded; that is, there is a finite upper bound for the radii of balls contained in C.

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.

A Note on Best Simultaneous Approximation in Normed Linear Spaces

1976 ◽  
Vol 19 (3) ◽  
pp. 359-360 ◽  
Author(s):  
Arne Brøndsted
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Convex Functions ◽  
Existence And Uniqueness ◽  
Simultaneous Approximation ◽  
Normed Linear Space ◽  
Present Note ◽  
Linear Spaces ◽  
Best Simultaneous Approximation ◽  
Image Position

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that

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