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… (More)