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IIT Hyderabad

We continue to study (strong) property-(R 1) in Banach spaces. As discussed by Pai &amp; Nowroji in [On restricted centers of sets, J. Approx. Theory, 66(2), 170–189 (1991)], this study corresponds to a triplet (Figure presented.), where X is a Banach space, V is a closed convex set, and (Figure presented.) is a subfamily of closed, bounded subsets of X. It is observed that if X is a Lindenstrauss space then (Figure presented.) has strong property-(R 1), where (Figure presented.) represents the compact subsets of X. It is established that for any (Figure presented.). This extends the well-known fact that a compact subset of a Lindenstrauss space X admits a nonempty Chebyshev center in X. We extend our observation that (Figure presented.) is Lipschitz continuous in (Figure presented.) if X is a Lindenstrauss space. If Y is a subspace of a Banach space X and (Figure presented.) represents the set of all finite subsets of BX then we observe that BY exhibits the condition for simultaneously strongly proximinal (viz. property-(P 1)) in X for (Figure presented.) if (Figure presented.) satisfies strong property-(R 1), where (Figure presented.) represents the set of all finite subsets of X. It is demonstrated that if P is a bi-contractive projection in ℓ ∞, then (Figure presented.) exhibits the strong property-(R 1), where (Figure presented.) represents the set of all compact subsets of ℓ ∞. Furthermore, stability results for these properties are derived in continuous function spaces, which are then studied for various sums in Banach spaces. © 2023 NISC (Pty) Ltd.

World University root node

Department of Mathematics

Fulltext

Quaestiones Mathematicae

<a href="https://www.scopus.com/record/display.uri?eid=2-s2.0-85165467960&amp;origin=resultslist&amp;sort=cp-f&amp;src=s&amp;st1=Paul&amp;st2=Tanmoy&amp;nlo=1&amp;nlr=20&amp;nls=count-f&amp;sid=fee942724b111309d71d67a466a28e0e&amp;sot=anl&amp;sdt=aut&amp;sl=29&amp;s=AU-ID%28%22Paul%2c+T.%22+56083572600%29&amp;relpos=9&amp;citeCnt=0&amp;searchTerm=">On property-(R 1) and relative Chebyshev centers in Banach spaces-II</a>

We demonstrate that if J is an M-ideal in a Lindenstrauss space X, then X satisfies the famous Smith and Ward identity, that is, (Formula presented.) for finite subsets F of X. We introduce strong property- (Formula presented.) for a triplet (Formula presented.) where X is a Banach space, V is a closed convex subset of X, and (Formula presented.) is a subfamily of closed, bounded subsets of X. We show that for a subspace V, the restricted Chebyshev centre for (Formula presented.) with respect to the unit ball of V is non-empty if the triplet (Formula presented.) has the strong property- (Formula presented.) We demonstrate that for an M-ideal J in a Lindenstrauss space X, the triplet (Formula presented.) has the strong property- (Formula presented.) where (Formula presented.) is the family of compact subsets of X. Some characterizations of the strong property- (Formula presented.) are given. Similar to the strong (Formula presented.) -ball property, we show that for a subspace V of X, a triplet (Formula presented.) has the strong property- (Formula presented.) if and only if (Formula presented.) has the property- (Formula presented.) and the set of restricted Chebyshev centres with respect to the unit ball of V is non-empty for (Formula presented.). © 2022 Taylor & Francis Group, LLC.

Numerical Functional Analysis and Optimization

On Property- and Relative Chebyshev Centres in Banach Spaces

In this paper, we analyze the various strengthening and weakening of the uniqueness of the Hahn–Banach extension. In addition, we consider the case in which Y is an ideal of X. In this context, we study the property (U)/(SU)/(HB) and property (wU)/(k-U) for a subspace Y of a Banach space X. We obtain various new characterizations of these properties. We study different kinds of stabilities resulting from these properties in the tensor product spaces, spaces of Bochner integrable functions, and the higher duals of Banach spaces. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is established that a hyperplane in c has property (HB) if and only if it is an M-summand. It is observed that a finite-dimensional subspace Y has property (k-U) in c, in addition to that if Y is an ideal, then Y∗ is a k-strictly convex subspace of ℓ1 for some natural k. © 2022, Tusi Mathematical Research Group (TMRG).

Advances in Operator Theory

Uniqueness of Hahn–Banach extensions and some of its variants

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in (Formula presented.) and when this association forms a linear operator of norm-1 from (Formula presented.) to (Formula presented.). It is proved that, under certain geometric assumptions on (Formula presented.) these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if (Formula presented.) has property-U (SU) in (Formula presented.). We prove that when (Formula presented.) has the Radon–Nikod (Formula presented.) m Property for (Formula presented.), (Formula presented.) has property-U (property-SU) in (Formula presented.) if and only if Y is so in X. We show that if (Formula presented.) and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and (Formula presented.) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in (Formula presented.) which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of (Formula presented.) which have property-U (SU). © 2021 Informa UK Limited, trading as Taylor & Francis Group.

Linear and Multilinear Algebra

Stability of unique Hahn–Banach extensions and associated linear projections

In this paper we characterize Property-(R1 ), a generalization of 11 ball property. As a necessary and sufficient condition of a subspace Y 2 with Property-(R1 ) we derive that r(y, F) = radY (F ) + d(y, centY (F )) for any bounded subset F and y ∈ Y . We introduce the notion of modulus of relative chebyshev centre and characterize Property-(R1 ) in terms of this modulus. It is observed that if Y is a finite co-dimensional strongly proximinal subspace of a L1 predual space X and F is a finite subset of X then radY (F ) = radX (F ) + d(F, Y ). We characterize continuity of centV (.) in terms of the modulus of relative chebyshev centre. © 2019, American Mathematical Society. All rights reserved.

Contemporary Mathematics

Some geometric properties of relative chebyshev centres in banach spaces

We show that a separable proximinal subspace of X, say Y is strongly proximinal (strongly ball proximinal) if and only if Lp(I, Y) is strongly proximinal (strongly ball proximinal) in Lp(I, X), for 1 ≤ p &lt; ∞. The p = ∞ case requires a stronger assumption, that of 'uniform proximinality'. Further, we show that a separable subspace Y is ball proximinal in X if and only if Lp(I, Y) is ball proximinal in Lp(I, X) for 1 ≤ p ≤ ∞. We develop the notion of 'uniform proximinality' of a closed convex set in a Banach space, rectifying one that was defined in a recent paper by P.-K Lin et al. [J. Approx. Theory 183 (2014), 72-81]. We also provide several examples having this property; viz. any U-subspace of a Banach space has this property. Recall the notion of 3.2.I.P. by Joram Lindenstrauss, a Banach space X is said to have 3.2.I.P. if any three closed balls which are pairwise intersecting actually intersect in X. It is proved the closed unit ball BX of a space with 3.2.I.P and closed unit ball of any M-ideal of a space with 3.2.I.P. are uniformly proximinal. A new class of examples are given having this property. © 2016 by the Tusi Mathematical Research Group.

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

In this paper, we study two properties (P1) and (P2) introduced in the literature for studying the existence and stability of relative Chebyshev centers. We reformulate these properties in a way that leads naturally to our results. We mainly relate these properties with some geometric properties of Banach spaces. In particular, reflexive spaces having the Kadec–Klee property are characterized in terms of property (P1) and uniform convexity is characterized in terms of property (P2). Some continuity results for the center map are also presented. © 2016 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Chebyshev centers and some geometric properties of Banach spaces

We investigate a variation of the transitivity problem for proximinality properties of subspaces and intersection properties of balls in Banach spaces. For instance, we prove that if Z⊆Y⊆X, where Z is a finite co-dimensional subspace of X which is strongly proximinal in Y and Y is an M-ideal in X, then Z is strongly proximinal in X. Towards this, we prove that a finite co-dimensional proximinal subspace Y of X is strongly proximinal in X if and only if Y⊥⊥ is strongly proximinal in X**. We also prove that in an abstract L1-space, the notions of strongly subdifferentiable points and quasi-polyhedral points coincide. We also give an example to show that M-ideals need not be ball proximinal. Moreover, we prove that in an L1-predual space, M-ideals are ball proximinal. © 2015 Elsevier Inc.

Strong proximinality and intersection properties of balls in Banach spaces

We investigate almost ball remotal and ball remotal subspaces of Banach spaces. Several subspaces of the classical Banach spaces are identified having these properties. Some stability results for these properties are also proved. © Instytut Matematyczny PAN, 2012.

Studia Mathematica

Almost ball remotal subspaces in Banach spaces

Positivity

Ball remotality of M-ideals in some function spaces and function algebras

In this paper, we continue our study of ball remotality of subspaces of Banach spaces. In particular, we study the problem in classical sequence spaces c0, c, l1 and l∞; and also ball remotality of a Banach space in its bidual. © 2010 Elsevier Inc.

Ball remotality in Banach spaces

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)**. © 2009.