On Property- and Relative Chebyshev Centres in Banach Spaces. (12th March 2022) - Daptari, Soumitra; Paul, Tanmoy
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, rad J ( F ) = rad X ( F ) + d ( cent X ( F ), J ) for finite subsets F of X . We introduce strong property- ( R 1 ) for a triplet ( X, V, F ), where X is a Banach space, V is a closed convex subset of X, and F is a subfamily of closed, bounded subsets of X . We show that for a subspace V, the restricted Chebyshev centre for F ∈ F with respect to the unit ball of V is non-empty if the triplet ( X, V, F ) has the strong property- ( R 1 ) . We demonstrate that for an M -ideal J in a Lindenstrauss space X, the triplet ( X, J, K ( X ) ) has the strong property- ( R 1 ), where 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 1 2 -ball property, we show that for a subspace V of X, a triplet ( X, V, F ) has the strong property- ( R 1 ) if and only if ( X, V, 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 ∈ F .