Best approximation in L(μ, X), II

The object of this paper is to prove the following theorem: If Y is a closed subspace of the Banach space X, then L¹(μ, Y) is proximinal in L¹(μ, X) if and only if L^p(μ, Y) is proximinal in L^p(μ, Y) for every p, 1 < p < ∞. As an application of this result we prove that if Y is either reflexive or Y is a separable proximinal dual space, then L¹(μ, Y) is proximinal in L¹(μ, X).