Best approximation in polyhedral Banach spaces Original Research Article

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