On proximinal subspaces of vector-valued Orlicz–Musielak spaces

We prove the following theorem: Let YY be a weakly KK-analytic subspace of a real Banach space XX, ΩΩ a complete σσ-finite measure space and Φ:Ω×R+→R+Φ:Ω×R+→R+ a φφ-function with a parameter which is strictly increasing with regard to its second variable. Then LΦ(Ω,Y)LΦ(Ω,Y) is proximinal in LΦ(Ω,X)LΦ(Ω,X) if and only if YY is proximinal in XX. This extends the result given for LpLp-spaces by Cascales and Raja. The Chebyshevity of the spaces LΦ(Ω,Y)LΦ(Ω,Y) and EΦ(Ω,Y)EΦ(Ω,Y) is also discussed.