Strong proximinality and intersection properties of balls in Banach spaces

We investigate a variation of the transitivity problem for proximinality properties of subspaces and intersection properties of balls in Banach spaces. For instance, we prove that if Z⊆Y⊆XZ⊆Y⊆X, where Z is a finite co-dimensional subspace of X which is strongly proximinal in Y and Y is an M-ideal in X, then Z is strongly proximinal in X. Towards this, we prove that a finite co-dimensional proximinal subspace Y of X is strongly proximinal in X   if and only if Y⊥⊥Y⊥⊥ is strongly proximinal in X⁎⁎X⁎⁎. We also prove that in an abstract L1L1-space, the notions of strongly subdifferentiable points and quasi-polyhedral points coincide. We also give an example to show that M  -ideals need not be ball proximinal. Moreover, we prove that in an L1L1-predual space, M-ideals are ball proximinal.