Best simultaneous approximation in L1(μ,X)

Let X be a Banach space, (Ω,Σ,μ) a finite measure space, and L1(μ,X) the Banach space of X-valued Bochner μ-integrable functions defined on Ω endowed with its usual norm. Let us suppose that Σ0 is a sub-σ-algebra of Σ, and let μ0 be the restriction of μ to Σ0. Given a natural number n, let N be a monotonous norm in Rn. It is shown that if X is reflexive then L1(μ0,X) is N-simultaneously proximinal in L1(μ,X) in the sense of Fathi et al. [Best simultaneous approximation in Lp(I,E), J. Approx. Theory 116 (2002), 369–379]. Some examples and remarks related with N-simultaneous proximinality are also given.