Some Geometric and Proximinality Properties in Banach Spaces

We present some geometric and approximation theoretic characterizations of locally uniform convex, compactly locally uniformly convex, nearly uniformly convex, compactly uniformly convex and uniformly convex Banach spaces. The approximation theoretic characterizations of compactly locally uniformly convex spaces presented in this paper provide some answers to a question posed by F. Deutsch and J. Lambert in their paper On continuity of metric projections, J. Approx. Theory 29 (1980) 116--131.