Porosity of mutually nearest and mutually furthest points in Banach spaces

Li, Chong; Xu, Hong–Kun

doi:10.1016/j.jat.2003.07.001

Cited by 13 publications

(7 citation statements)

References 13 publications

(23 reference statements)

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“…These results are in the spirit of the idea due to Blasi, Myjak and Papini in [7]. Extensions to convex sets and generalized approximations of this idea of Blasi, Myjak and Papini can be found in [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 58%

Existence and porosity for a class of perturbed optimization problems in Banach spaces

2007

Journal of Mathematical Analysis and Applications

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Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem sup z∈Z {J (z) + x − z }, which is denoted by (x, J )-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x ∈ X for which the problem (x, J )-sup has a solution is a dense G δ -subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist z 0 ∈ Z such that J (z 0 ) + x − z 0 = sup z∈Z {J (z) + x − z } is a σ -porous subset of X and the set of all points x ∈ X \ Z 0 such that there exists a maximizing sequence of the problem (x, J )-sup which has no convergent subsequence is a σ -porous subset of X \ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z is in the solution set of (z, J )-sup.

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Section: Introductionmentioning

confidence: 58%

Existence and porosity for a class of perturbed optimization problems in Banach spaces

2007

Journal of Mathematical Analysis and Applications

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“…The definition of a porous set given above, which is taken from [5,17], is stronger than the concept with the same name introduced by Zajíček in metric spaces (cf., e.g., [23]; the notion from Definition 2.4 is called there "uniformly very porous").…”

Section: Porositymentioning

confidence: 99%

“…As will be seen, such an extension is nontrivial. Extensions to convex sets and generalized approximations of the idea of Blasi, Myjak and Papini can be found in [4,[15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Porosity of perturbed optimization problems in Banach spaces

2006

Journal of Mathematical Analysis and Applications

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Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z → R be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem inf z∈Z {J (z) + x − z }, denoted by (x, J )-inf for x ∈ X. In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x in X for which there does not exist z 0 ∈ Z such that J (z 0 ) + x − z 0 = inf z∈Z {J (z) + x − z } is a σ -porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points x ∈ X \ Z 0 such that the problem (x, J )-inf fails to be approximately compact, is a σ -porous set in X \ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z ∈ P Z (z). Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417-424] fail is provided.

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“…For more developments and extensions in this direction, the readers are referred to [2,3,7,8,11,[22][23][24][25][26][27][28][29][31][32][33] and the surveys [12,30].…”

Section: Introductionmentioning

confidence: 99%

Nearest and farthest points in spaces of curvature bounded below

Espínola

López-Acedo

2010

Journal of Approximation Theory

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Let A be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space (X, d) and x ∈ X \ A. The nearest point problem (resp. the farthest point problem) w.r.t. x considered here is to find a point a 0 ∈ A such that d(x, a 0 ) = inf{d(x, a) : a ∈ A} (resp. d(x, a 0 ) = sup{d(x, a) : a ∈ A}). We study the well posedness of nearest point problems and farthest point problems in geodesic spaces. We show that if X is a space of curvature bounded below then the complement of the set of all points x ∈ X for which the nearest point problem (resp. the farthest point problem) w.r.t. x is well posed is σ -porous in X \ A. Our results extend and/or improve some recent results about proximinality in geodesic spaces as well as the corresponding ones previously obtained in uniformly convex Banach spaces. In particular, the result regarding the nearest point problem answers affirmatively an open problem proposed by Kaewcharoen and Kirk [A. Kaewcharoen, W.A. Kirk, Proximinality in geodesic spaces, Abstr.

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Porosity of mutually nearest and mutually furthest points in Banach spaces

Cited by 13 publications

References 13 publications

Existence and porosity for a class of perturbed optimization problems in Banach spaces

Existence and porosity for a class of perturbed optimization problems in Banach spaces

Porosity of perturbed optimization problems in Banach spaces

Nearest and farthest points in spaces of curvature bounded below

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