Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

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.

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Orlicz Spaces Without Strongly Extreme Points and Without H-Points

Canadian Mathematical Bulletin ◽

10.4153/cmb-1993-025-6 ◽

1993 ◽

Vol 36 (2) ◽

pp. 173-177 ◽

Cited By ~ 4

Author(s):

Henryk Hudzik

Keyword(s):

Extreme Point ◽

Orlicz Space ◽

Unit Sphere ◽

Orlicz Function ◽

Orlicz Spaces ◽

Extreme Points ◽

Luxemburg Norm ◽

Nonatomic Measure

AbstractW. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

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Extreme and Exposed Points in Orlicz Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1992-032-3 ◽

1992 ◽

Vol 44 (3) ◽

pp. 505-515 ◽

Cited By ~ 7

Author(s):

R. Grzaślewicz ◽

H. Hudzik ◽

W. Kurc

Keyword(s):

Orlicz Space ◽

Unit Sphere ◽

Measure Space ◽

Orlicz Function ◽

Orlicz Spaces ◽

Extreme Points ◽

Infinite Measure ◽

Nonatomic Measure

AbstractExtreme points of the unit sphere in any Orlicz space over a measure space that contains no atoms of infinite measure are characterized. In the case of a finite-valued Orlicz function and a nonatomic measure space, exposed points of the unit sphere in these spaces are characterized too. Some corollaries and examples are also given.

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

Publications de l Institut Mathematique ◽

10.2298/pim1409229s ◽

2014 ◽

Vol 95 (109) ◽

pp. 229-238 ◽

Cited By ~ 2

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.

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Smooth Norms in Orlicz Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1991-012-7 ◽

1991 ◽

Vol 34 (1) ◽

pp. 74-82 ◽

Cited By ~ 5

Author(s):

R. P. Maleev ◽

S. L. Troyanski

Keyword(s):

Orlicz Spaces ◽

Equivalent Norm ◽

Equivalent Norms ◽

Fréchet Differentiability ◽

Frechet Differentiability ◽

Fréchet Differentiable

AbstractEquivalent norms with best order of Frechet and uniformly Frechet differentiability in Orlicz spaces are constructed. Classes of Orlicz which admit infinitely many times Frechet differentiable equivalent norm are found.

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Convexities and Existence of the Farthest Point

Abstract and Applied Analysis ◽

10.1155/2011/139597 ◽

2011 ◽

Vol 2011 ◽

pp. 1-9 ◽

Cited By ~ 2

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.

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