Relative Chebyshev Centers in L_(∞)(μ,X)

Let $X$ be a Banach space and $Y$ a weakly $\mathcal{K}$-analytic subspace of $X$. In this paper we study the simultaneous proximinality in the space $L_\infty(\mu,X)$. In this sense we have proved that $L_\infty(\mu,Y)$ is simultaneously proximinal in $L_\infty(\mu,X)$ if, and only if, $Y$ is simultaneously proximinal in $X$