Almost Chebyshev set with respect to bounded subsets

Li, Chong; Wang, Xinghua

doi:10.1007/bf02911437

Cited by 10 publications

(4 citation statements)

References 10 publications

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“…Some other generic results in spaces of convex sets can be found in [3,5,12,15]. The results of the present paper generalize and sharpen some results from [4,5,12], etc.…”

Section: Introductionsupporting

confidence: 73%

Porosity of mutually nearest and mutually furthest points in Banach spaces

2003

Journal of Approximation Theory

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Let X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X : Let BðX Þ (resp. KðX Þ) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let B G ðX Þ denote the closure of the set fAABðX Þ : A-G ¼ |g and K G ðX Þ ¼ B G ðX Þ-KðX Þ: We introduce the admissible family A of BðX Þ and prove that E o A ðGÞ (resp. E A o ðGÞ), the set of all subsets F AADB G ðX Þ (resp. F AADBðX Þ) such that the minimization problem minðF ; GÞ (resp. the maximization problem maxðF ; GÞ) is well-posed, is a dense G d -subset of A: Furthermore, when X is uniformly convex, we prove that A\E o A ðGÞ and A\E A o ðGÞ are s-porous in A: r

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“…Some other generic results in spaces of convex sets can be found in [3,5,12,15]. The results of the present paper generalize and sharpen some results from [4,5,12], etc.…”

Section: Introductionsupporting

confidence: 73%

Porosity of mutually nearest and mutually furthest points in Banach spaces

2003

Journal of Approximation Theory

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“…Some further developments of Stechkin's ideas can be founded in [2 6, 8, 11 17, 20, 24, 26] and in the monograph [10], while some generic results in spaces of convex sets and bounded sets can be founded in [2,3,7,19,21].…”

Section: K(x)=[a # C(x) : a Is Compact]mentioning

confidence: 99%

On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces

2000

Journal of Approximation Theory

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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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Almost Chebyshev set with respect to bounded subsets

Cited by 10 publications

References 10 publications

Porosity of mutually nearest and mutually furthest points in Banach spaces

Porosity of mutually nearest and mutually furthest points in Banach spaces

On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces

Nearest and farthest points in spaces of curvature bounded below

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