Abstract: In this paper, we consider the best simultaneous approximations in Bochner-Lebesgue spaces with respective to Minkowski' norms in Euclidean spaces. Firstly, we give a characterization of best simultaneous approximations by the distance functions. Then, by applying this characterization and a measurable selection theorem we show the simultaneous proximinality of a Bochner-Lebesgue space whose functions take values in a closed separable subspace is equivalent to the simultaneous proximinality of the closed separable subspace. Finally, we conclude that for their equivalence, the separability of the subspace is necessary.

Key words: Bochner-Lebesgue space, simultaneously proximinality, best simultaneous approximation