On Property- and Relative Chebyshev Centres in Banach Spaces

Abstract

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, ${\text{rad}}_J(F)={\text{rad}}_X(F)+d({\text{cent}}_X(F),J)$ for finite subsets F of X. We introduce strong property-$(R_1)$ for a triplet $(X,V,\mathcal{F}),$ where X is a Banach space, V is a closed convex subset of X, and $\mathcal{F}$ is a subfamily of closed, bounded subsets of X. We show that for a subspace V, the restricted Chebyshev centre for $F\in \mathcal{F}$ with respect to the unit ball of V is non-empty if the triplet $(X,V,\mathcal{F})$ has the strong property-$(R_1).$ We demonstrate that for an M-ideal J in a Lindenstrauss space X, the triplet $(X,J,\mathcal{K}(X))$ has the strong property-$(R_1),$ where $\mathcal{K}(X)$ is the family of compact subsets of X. Some characterizations of the strong property-$(R_1)$ are given. Similar to the strong $1\frac{1}{2}$-ball property, we show that for a subspace V of X, a triplet $(X,V,\mathcal{F})$ has the strong property-$(R_1)$ if and only if $(X,V,\mathcal{F})$ has the property-$(R_1)$ and the set of restricted Chebyshev centres with respect to the unit ball of V is non-empty for $F\in \mathcal{F}.$

Keywords:

• $1\frac{1}{2}$-ball property
• Chebyshev centre
• Lindenstrauss space
• M-ideal
• property-$(R_1)$
• restricted Chebyshev centre

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 46B20
• 41A65

Acknowledgement

The authors would like to express their gratitude to the referees for their thorough evaluation of the manuscript, which improved its readability and interpretation.

Additional information

Funding

The second author’s research was supported by the SERB, India. Award No. MTR/2017/000061.