Strong ball proximinality and continuity of metric projection in L1-predual spaces

In this article, we discuss the strong proximinality of the closed unit ball of closed linear subspaces of L1-predual spaces. We prove that M-ideals in L1-predual spaces are strongly ball proximinal. We characterize finite co-dimensional strongly ball proximinal closed linear subspaces of C(K) spaces and prove that they are precisely the finite co-dimensional strongly proximinal closed linear subspaces of C(K). For a Choquet simplex K, we give a sufficient condition for the strong ball proximinality of finite co-dimensional closed linear subspaces of A(K) spaces. We prove that metric projection onto a finite co-dimensional strongly proximinal closed linear subspace of an L1-predual space is Hausdorff metric continuous. Moreover, we prove that the metric projection onto the closed unit ball of an M-ideal in an L1-predual space is Hausdorff metric continuous.