Best approximation in polyhedral Banach spaces

Abstract

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