Existence of best simultaneous approximations in Lp(S, Σ, X)

Let (S, Sigma, mu) be a complete positive sigma-finite measure space and let X be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in L-p(S,Sigma, X). Let Sigma(0) be a sub-sigma-algebra of Sigma and Y a nonempty locally weakly compact convex subset of X such that (span Y) over bar and its dual have the Radon-Nikodym property. We prove that L-p(S, Sigma(0), Y) is N-simultaneous proximinal in L-p(S, Sigma, X) (with the additional assumption that (S, Sigma, mu) be finite for the case when p = 1). Furthermore, for the special case when Sigma(0) = Sigma, we show that the assumption that the dual of (span Y ) over bar has the Radon-Nikodym property can be removed. (C) 2011 Elsevier Inc. All rights reserved.