Proximinality in Banach spaces ✩

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