Existence of best simultaneous approximations in Lp(S,Σ,X) without the RNP assumption

Let ( S ,Σ, µ) be a complete positive σ -finite measure space and let X be a Banach space. We consider the simultaneous proximinality problem in L p ( S ,Σ, X ) for 1 ⩽ p < +∞. We establish some N -simultaneous proximinality results of L p ( S ,Σ 0 , Y ) in L p ( S ,Σ, X ) without the Radon-Nikodým property (RNP) assumptions on the space and its dual , where Σ 0 is a sub- σ -algebra of Σ and Y a nonempty locally weakly compact closed convex subset of X . In particular, we completely solve one open problem and partially solve another one in Luo et al. (2011).