Stability of ball proximinality

In this paper, we show that if E is an order continuous Köthe function space and Y is a separable subspace of X, then E(Y) is ball proximinal in E(X) if and only if Y is ball proximinal in X. As a consequence, E(Y) is proximinal in E(X) if and only if Y is proximinal in X. This solves an open problem of Bandyopadhyay, Lin and Rao. It is also shown that if E is a Banach lattice with a 1-unconditional basis and for each n, Yn is a subspace of Xn, then (⊕Yn)E is ball proximinal in (⊕Xn)E if and only if each Yn is ball proximinal in Xn.