  In this article, we prove the proximinality of closed unit ball of $M$-ideals
of compact operators. We also prove the ball proximinality of $M$-embedded
spaces in their biduals. Moreover, we show that $\mathcal{K}(\ell_1)$, the
space of compact operators on $\ell_1$, is ball proximinal in
$\mathcal{B}(\ell_1)$, the space of bounded operators on $\ell_1$, even though
$\mathcal{K}(\ell_1)$ is not an $M$-ideal in $\mathcal{B}(\ell_1)$.

Functional Analysis