scholarly journals On the farthest points in convex metric spaces and linear metric spaces

2014 ◽  
Vol 95 (109) ◽  
pp. 229-238 ◽  
Author(s):  
S Sangeeta ◽  
T.D. Narang
Keyword(s):  
Metric Spaces ◽  
Farthest Points ◽  
Strictly Convex ◽  
Convex Metric Spaces ◽  
Farthest Point ◽  
Farthest Point Map

We prove some results on the farthest points in convex metric spaces and in linear metric spaces. The continuity of the farthest point map and characterization of strictly convex linear metric spaces in terms of farthest points are also discussed.

scholarly journals Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
H. R. Khademzadeh ◽  
H. Mazaheri
Keyword(s):  
Orlicz Function ◽  
Orlicz Spaces ◽  
Farthest Points ◽  
Strong Solvability ◽  
Fréchet Differentiability ◽  
Measure Spaces ◽  
Nonatomic Measure ◽  
Frechet Differentiability ◽  
Farthest Point ◽  
Farthest Point Map

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such thatXbecomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spacesL ​ϕ(μ)andE ​ϕ(μ)over nonatomic measure spaces in terms of the functionϕ. We will prove that the Fréchet differentiability of the farthest point map and the conditionsϕ∈Δ2andϕ>0in reflexive Musielak-Orlicz function spaces are equivalent.

scholarly journals Convexities and Existence of the Farthest Point

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Z. H. Zhang ◽  
C. Y. Liu
Keyword(s):  
Banach Space ◽  
Closed Subset ◽  
Sufficient Condition ◽  
Residual Subset ◽  
Farthest Points ◽  
Weakly Closed ◽  
Farthest Point ◽  
Farthest Point Map

Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of when is very convex.

scholarly journals An Intrinsic Characterization of Five Points in a CAT(0) Space

2020 ◽  
Vol 8 (1) ◽  
pp. 114-165
Author(s):  
Tetsu Toyoda
Keyword(s):  
Metric Space ◽  
Isometric Embedding ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
New Approach ◽  
Intrinsic Characterization ◽  
New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

RANDOM COINCIDENCE AND FIXED POINTS FOR WEAKLY COMPATIBLE MAPPINGS IN CONVEX METRIC SPACES

2009 ◽  
Vol 02 (02) ◽  
pp. 171-182 ◽  
Author(s):  
Izmat Beg ◽  
Adnan Jahangir ◽  
Akbar Azam
Keyword(s):  
Fixed Points ◽  
Metric Spaces ◽  
Weakly Compatible Mappings ◽  
Compatible Mappings ◽  
Coincidence Points ◽  
Random Coincidence ◽  
Complete Metric Spaces ◽  
Random Fixed Points ◽  
Weakly Compatible ◽  
Convex Metric Spaces

Some new theorems on random coincidence points and random fixed points for weakly compatible mappings in convex separable complete metric spaces have been established. These results generalize some recent well known comparable results in the literature.

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