2021/08/02 by Thomas, Teena
#41A50(Primary) 52A07 #41A65 #46E15(Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)
paper · doi:10.48550/arxiv.2108.00628
We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by J. Mach [Continuity properties of Chebyshev centers, J. Approx. Theory, 29(3):223--230, 1980] as property-(P1). We establish that if the closed unit ball of a closed subspace of a Banach space X possesses property-(P1) for each of the classes of closed bounded, compact and finite subsets of X, then so does the subspace. It is also proved that the closed unit ball of an M-ideal in an L1-predual space satisfies property-(P1) for the compact subsets of the space. For a Choquet simplex K, we provide a sufficient condition for the closed unit ball of a finite co-dimensional closed subspace of A(K) to satisfy property-(P1) for the compact subsets of A(K). This condition also helps to establish the equivalence of strong proximinality of the closed unit ball of a finite co-dimensional subspace of A(K) and property-(P1) of the closed unit ball of the subspace for the compact subsets of A(K). Further, for a compact Hausdorff space S, a characterization is provided for a strongly proximinal finite co-dimensional closed subspace of C(S) in terms of property-(P1) of the subspace and that of its closed unit ball for the compact subsets of C(S). We generalize this characterization for a strongly proximinal finite co-dimensional closed subspace of an L1-predual space. As a consequence, we prove that such a subspace is a finite intersection of hyperplanes such that the closed unit ball of each of these hyperplanes satisfy property-(P1) for the compact subsets of the L1-predual space and vice versa. We conclude this article by providing an example of a closed subspace of a non-reflexive Banach space which satisfies 1 (1)/(2)-ball property and does not admit restricted Chebyshev centre for a closed bounded subset of the Banach space.