Abstract

D. Yost [ Intersecting balls and proximinal subspaces in Banach spaces, Ph.D. Thesis, University of Edinburgh (1979)] introduced a notion called equability for subspaces of a Banach space, as a sufficient condition for the proximinality and lower semi-continuity of the metric projection. This notion was later modified by S. Lalithambigai [ Ball proximinality of equable spaces, Collect. Math. 60/1 (2009) 79–88] to study ball proximinality. In this article, we study both the equability notions. We establish some geometric characterizations and strengthen the existing best approximation theoretic consequences of equability notions. Specifically, we prove that these equability notions are related to quasi uniform rotundity and uniform rotundity. We observe that equability notions are transitive through M-summands and further explore their stability under ℓp\ell_p-direct sum for 1 ≤ p ≤ ∞. Additionally, we show that equability can be lifted to the space of all vector valued bounded (and continuous) functions defined on topological spaces.

Author information

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

P. Gayathri

Dept. Mathematical and Computational Sciences, National Institute of Technology Karnataka,
Surathkal, Mangaluru, India

pgayathri@nitk.edu.in

V. Thota

Dept. of Mathematics, National Institute of Technology, Tiruchirappalli, Tamil Nadu, India

vamsinadh@nitt.edu

Keywords

  • Equability
  • strong equability
  • quasi uniform rotundity
  • uniform rotundity
  • uniform strong proximinality
  • stability

Mathematics Subject Classification

41A65, 46B20; 46E40

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Suggested citation

P. Gayathri, K. Sreelakshmi, V. Thota. “Equability and Strong Equability in Banach Spaces.” Journal of Convex Analysis 33 (2026), No. 1&2, 521–541. https://doi.org/10.68381/jca33033

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.