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

Let (S,Σ , μ) be a complete positive σ -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,Σ , X). Let Σ0 be a sub-σ -algebra of Σ and Y a nonempty locally weakly compact convex subset of X such that span Y and its dual have the Radon–Nikodym property. We prove that L p(S,Σ0, Y ) is N -simultaneous proximinal in L p(S,Σ , X) (with the additional assumption that (S,Σ , μ) be finite for the case when p = 1). Furthermore, for the special case when Σ0 = Σ , we show that the assumption that the dual of span Y has the Radon–Nikodym property can be removed. c ⃝ 2011 Elsevier Inc. All rights reserved.