On restricted centers of sets

Pai, D. V.; Nowroji, P. T.

doi:10.1016/0021-9045(91)90119-u

Cited by 20 publications

(7 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We adopt the idea introduced in [19] in terms of Property-(R 2 ) and define 'uniform proximinality' of a closed convex set. Section 4 is devoted to discussing this property.…”

Section: Main Results and Subsequent Discussionmentioning

confidence: 99%

“…We now define a stronger version of proximinality, viz. uniform proximinality which is in fact stated in [19] in the context of centres of closed bounded sets. Definition 4.2.…”

Section: Uniform Proximinality Of Lmentioning

confidence: 97%

“…We refer [19] to the reader for many other interesting uniformly proximinal subsets of Banach spaces. (c) From the example by Godefroy in [11] it is clear that the closed unit ball of a Banach space not necessarily have uniformly proximinal property.…”

Section: Uniform Proximinality Of Lmentioning

confidence: 99%

See 2 more Smart Citations

Various notions of best approximation property in spaces of Bochner integrable functions

Paul

2016

Preprint

View full text Add to dashboard Cite

We derive that for a separable proximinal subspace Y of X, Y is strongly proximinal (strongly ball proximinal) if and only if for 1 ≤ p < ∞, Lp(I, Y ) is strongly proximinal (strongly ball proximinal) in Lp(I, X). Case for p = ∞ follows from stronger assumption on Y in X (uniform proximinality). It is observed that for a separable proximinal subspace Y in X, Y is ball proximinal in X if and only if Lp(I, Y ) is ball proximinal in Lp(I, X) for 1 ≤ p ≤ ∞. Our observations also include the fact that for any (strongly) proximinal subspace Y of X, if every separable subspace of Y is ball (strongly) proximinal in X then Lp(I, Y ) is ball (strongly) proximinal in Lp(I, X) for 1 ≤ p < ∞. We introduce the notion of uniform proximinality of a closed convex set in a Banach space, which is wrongly defined in [16]. Several examples are given having this property, viz. any U -subspace of a Banach space, closed unit ball BX of a space with 3.2.I.P , closed unit ball of any Mideal of a space with 3.2.I.P. are uniformly proximinal. A new class of examples are given having this property.

show abstract

“…We adopt the idea introduced in [19] in terms of Property-(R 2 ) and define 'uniform proximinality' of a closed convex set. Section 4 is devoted to discussing this property.…”

Section: Main Results and Subsequent Discussionmentioning

confidence: 99%

“…We now define a stronger version of proximinality, viz. uniform proximinality which is in fact stated in [19] in the context of centres of closed bounded sets. Definition 4.2.…”

Section: Uniform Proximinality Of Lmentioning

confidence: 97%

See 1 more Smart Citation

Various notions of best approximation property in spaces of Bochner integrable functions

Paul

2016

Preprint

View full text Add to dashboard Cite

show abstract

“…Furthermore, for δ ą 0, let cent V pB, δq " tv P V : rpv, Bq ď rad V pBq `δu. Definition 1.1 ( [13]). Let X be a Banach space, V P CVpXq and F Ď CBpXq.…”

Section: Introductionmentioning

confidence: 99%

Restricted Chebyshev centers in $L_1$-predual spaces

Thomas¹

2022

Preprint

View full text Add to dashboard Cite

In this paper, we provide a necessary and sufficient condition for the existence of a restricted Chebyshev center of a compact subset of an L 1 -predual space in a closed convex subset of the L 1 -predual space. We also provide a geometrical characterization of an L 1 -predual space in terms of the restricted Chebyshev radius in the following manner. A real Banach space X is an L 1 -predual space if and only if for each nonempty finite subset F of X and closed convex subset V of X, rad V pF q " rad X pF q `dpV, cent X pF qq, where we denote rad X pF q, rad V pF q, cent X pF q and dpV, cent X pF qq to be the Chebyshev radius of F in X, the restricted Chebyshev radius of F in V , the set of Chebyshev centers of F in X and the distance between the sets V and cent X pF q respectively. Furthermore, we explicitly describe the Chebyshev centers of closed bounded subsets of an M -summand in the space of real-valued continuous functions on a compact Hausdorff space.

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

On restricted centers of sets

Cited by 20 publications

References 14 publications

Various notions of best approximation property in spaces of Bochner integrable functions

Various notions of best approximation property in spaces of Bochner integrable functions

Restricted Chebyshev centers in $L_1$-predual spaces

Nearest and farthest points in spaces of curvature bounded below

Contact Info

Product

Resources

About