Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • k-strict convexity
  • k-strong convexity
  • k-uniform convexity
  • property k-UC
  • monotone norm
  • product space

Mathematics Subject Classification

46B20, 41A65

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Suggested citation

M. Veena Sangeetha, M. Radhakrishnan, S. Kar. “On k-Strong Convexity in Banach Spaces.” Journal of Convex Analysis 28 (2021), No. 4, 1193–1210. https://doi.org/10.68381/jca28070

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press.