Journal of Convex Analysis 24 (2017), No. 4, 1263--1279
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, U-proximinal, metric projection, uniform convexity.
MSC: 41A65, 46B20
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