Best approximation in polyhedral Banach spaces

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space XX onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if XX satisfies (∗)(∗) (a geometric property stronger than polyhedrality) and Y⊂XY⊂X is any proximinal subspace, then the metric projection PYPY is Hausdorff continuous and YY is strongly proximinal (i.e., if {yn}⊂Y{yn}⊂Y, x∈Xx∈X and ‖yn−x‖→dist(x,Y), then dist(yn,PY(x))→0).One of the main results of a different nature is the following: if XX satisfies (∗)(∗) and Y⊂XY⊂X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) YY is strongly proximinal; (b) YY is proximinal; (c) each element of Y⊥Y⊥ attains its norm. Moreover, in this case the quotient X/YX/Y is polyhedral.The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.