Daptari, Soumitra and Paul, Tanmoy
(2022)
On Property and Relative Chebyshev Centres in Banach Spaces.
Taylor and Francis Ltd..
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Abstract
We demonstrate that if J is an Mideal 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 nonempty if the triplet (Formula presented.) has the strong property (Formula presented.) We demonstrate that for an Mideal 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 nonempty for (Formula presented.). © 2022 Taylor & Francis Group, LLC.
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