On Property-$(P_{1})$ in Banach spaces

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-$(P_1)$. We establish that if the closed unit ball of a closed subspace of a Banach space $X$ possesses property-$(P_1)$ 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 $L_{1}$-predual space satisfies property-$(P_{1})$ 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-$(P_{1})$ 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-$(P_{1})$ 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-$(P_{1})$ 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 $L_{1}$-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-$(P_1)$ for the compact subsets of the $L_1$-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 $1 \frac{1}{2}$-ball property and does not admit restricted Chebyshev centre for a closed bounded subset of the Banach space.