Best approximation in polyhedral Banach spaces

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 82X is any proximinal subspace, then the metric projection PY is Hausdorff continuous and Y is strongly proximinal (i.e., if {yn} 82Y, x 08X and 25yn-x 25\u2192dist(x,Y), then dist(yn,PY(x))\u21920).One of the main results of a different nature is the following: if X satisfies (*) and Y 82X 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 a5 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