scholarly journals Characterisation of normed linear spaces with Mazur's intersection property

1978 ◽  
Vol 18 (1) ◽  
pp. 105-123 ◽  
Author(s):  
J.R. Giles ◽  
D.A. Gregory ◽  
Brailey Sims
Keyword(s):  
Euclidean Space ◽  
Convex Set ◽  
Intersection Property ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Strongly Exposed Points ◽  
Denting Points ◽  
Mazur's Intersection Property ◽  
Closed Convex Set ◽  
Space Property

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

Uniform Mazur's Intersection Property of Balls

1987 ◽  
Vol 30 (4) ◽  
pp. 455-460 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Convex Set ◽  
Closed Ball ◽  
Intersection Property ◽  
Mazur's Intersection Property ◽  
Closed Convex Set

AbstractWe give a dual characterization of the following uniformization of the Mazur's intersection property of balls in a Banach space X: for every ∊ > 0 there is a K > 0 such that whenever a closed convex set C ⊂ X and a point p ∊ X are such that diam C ≤ 1/∊ and dist(p, C) ≤ ∊, then there is a closed ball B of radius ≤ K with B ⊃ C and dist(p,B) ≥ ∊/2.

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.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

An orthogonality in normed linear spaces based on angular distance inequality

2015 ◽  
Vol 90 (2) ◽  
pp. 281-297 ◽  
Author(s):  
F. Dadipour ◽  
F. Sadeghi ◽  
A. Salemi
Keyword(s):  
Angular Distance ◽  
Linear Spaces ◽  
Normed Linear Spaces
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