Let be a subspace of a Banach space and, for , let denote the space of all -Bochner integrable functions on with values in . Let be a norm on such that whenever and . Then is said to be -simultaneously proximinal in if for each pair in there exists such that, for all ,
that is, is an -simultaneous best approximation to the pair . The main result of the paper is that if is reflexive and then is -simultaneously proximinal in .