Journal of Convex Analysis 28 (2021), No. 4, 1193--1210
Copyright Heldermann Verlag 2021
On k-Strong Convexity in Banach Spaces
M. Veena Sangeetha
Dept. of Mathematics, Indian Institute of Technology, Kharagpur, India
veena176@gmail.com
M. Radhakrishnan
Ramanujan Inst. for Advanced Study in Mathematics, University of Madras, India
and: Dept. of Mathematics, SRM University, Andra Pradesh, India
radhariasm@gmail.com
Samir Kar
Dept. of Mathematics, Indian Institute of Technology, Madras, India
and: Dept. of Mathematics, Indian Institute of Technology, Jammu, India
msamirkar@gmail.com
We introduce and study the notion of k-strong convexity in Banach spaces. It is a generalization of the notion of strong convexity first studied by Fan and Glicksberg. A Banach space is said to be k-strongly convex if it is reflexive, k-strictly convex and has the Kadec-Klee property. We use the idea of k-dimensional diameter to give several characterizations of k-strong convexity. Further, we study k-strict convexity and k-strong convexity in some products of Banach spaces. Finally, we give characterizations of k-uniform convexity that distinguish it from k-strong convexity.
Keywords: k-strict convexity, k-strong convexity, k-uniform convexity, property k-UC, monotone norm, product space.
MSC: 46B20, 41A65.
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