
Journal of Convex Analysis 25 (2018), No. 4, 11391158 Copyright Heldermann Verlag 2018 Some Geometric and Proximinality Properties in Banach Spaces P. Shunmugaraj Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India psraj@iitk.ac.in Vamsinadh Thota Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India vamst@iitk.ac.in 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) 116131. Keywords: Compactly locally uniformly convex, nearly uniformly convex, compactly uniformly convex, approximative compactness, metric projection. MSC: 41A65, 46B20 [ Fulltextpdf (141 KB)] for subscribers only. 