On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces

Li, Chong

doi:10.1006/jath.1999.3407

Cited by 18 publications

(12 citation statements)

References 21 publications

(17 reference statements)

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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%

“…Recently, the first author [12] of the present paper established the same results for compact subsets in reflexive locally uniformly convex Banach spaces.…”

Section: Introductionsupporting

confidence: 66%

“…Furthermore, when X is uniformly convex, we prove that the collection of all subsets in the admissible family such that the minimization (respectively, maximization) problems fail to be well-posed is s-porous in the admissible family. Applying the results to the admissible families BðX Þ and KðX Þ; we immediately extend some recent results due to De Blasi et al [5] and Li [12].…”

Section: Introductionmentioning

confidence: 53%

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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%

“…Recently, the first author [12] of the present paper established the same results for compact subsets in reflexive locally uniformly convex Banach spaces.…”

Section: Introductionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 53%

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Porosity of mutually nearest and mutually furthest points in Banach spaces

2003

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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“…Stečkin [23] was one of the first who realized that in case E is a Banach space, geometric properties like strict convexity, uniform convexity, reflexivity and others play an important role in the study of nearest and farthest point problems. His work triggered a series of results so-called "in the spirit of Stečkin" because the ideas he used were adapted again and again by different authors to various contexts (see, for example, [7,8,18]). In [23], Stečkin proved, in particular, that for each nonempty and closed subset A of a uniformly convex Banach space, the complement of the set of all points x ∈ E for which the nearest point problem of x to A has a unique solution is of first Baire category.…”

Section: Introductionmentioning

confidence: 99%

Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces

García

Nicolae

2010

Monatsh Math

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Let A and X be nonempty, bounded and closed subsets of a geodesic metric space (E, d). The minimization (resp. maximization) problem denoted by min(A, X) (resp. max(A, X)) consists in finding (a0,x0)∈A×X(a0,x0)∈A×X such that d(a0,x0)=inf{d(a,x):a∈A,x∈X}d(a0,x0)=inf{d(a,x):a∈A,x∈X} (resp. d(a0,x0)=sup{d(a,x):a∈A,x∈X}d(a0,x0)=sup{d(a,x):a∈A,x∈X}). We give generic results on the well-posedness of these problems in different geodesic spaces and under different conditions considering the set A fixed. Besides, we analyze the situations when one set or both sets are compact and prove some specific results for CAT(0) spaces. We also prove a variant of the Drop Theorem in Busemann convex geodesic spaces and apply it to obtain an optimization result for convex functions.Dirección General de Enseñanza SuperiorJunta de AntalucíaThe Sectoral Operational Programme Human Resources Developmen

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On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces

Cited by 18 publications

References 21 publications

Porosity of mutually nearest and mutually furthest points in Banach spaces

Porosity of mutually nearest and mutually furthest points in Banach spaces

Nearest and farthest points in spaces of curvature bounded below

Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces

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