
Journal of Convex Analysis 24 (2017), No. 4, 12631279 Copyright Heldermann Verlag 2017 Uniform Strong Proximinality and Continuity of Metric Projection Sudipta Dutta Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208108, India sudipta@iitk.ac.in P. Shunmugaraj Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208108, India psraj@iitk.ac.in Vamsinadh Thota Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208108, India vamst@iitk.ac.in We present a sufficient condition for the uniform continuity of metric projection. This condition is a natural strengthening of the notion of strong proximinality, appearing in the literature of the past few years. We show that this condition is equivalent to the sufficient condition of continuity of metric projection introduced by K.S. Lau back in 1979. A characterization of uniform convexity through proximinality is presented and we also relate quantitatively the power type estimate of modulus of uniform strong proximinality to the power type estimate of modulus of uniform convexity. Keywords: Uniformly strongly proximinal, Uproximinal, metric projection, uniform convexity. MSC: 41A65, 46B20 [ Fulltextpdf (131 KB)] for subscribers only. 