Proximinality in Banach spaces

In this paper, we study approximatively tau-compact and tau-strongly Chebyshev sets, where tau is the norm or the weak topology. We show that the metric projection onto tau-strongly Chebyshev sets are norm-tau continuous. We characterize approximatively tau-compact and tau-strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in tau-almost locally uniformly rotund spaces. We also prove some stability results on approximatively tau-compact and tau-strongly Chebyshev subspaces. (C) 2007 Elsevier Inc. All rights reserved.