Journal of Convex Analysis 33 (2026), No. 1&2, 521--541
Copyright Heldermann Verlag 2026
Equability and Strong Equability in Banach Spaces
P. Gayathri
Dept. Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangaluru, India
pgayathri@nitk.edu.in
K. Sreelakshmi
Dept. of Mathematics, National Institute of Technology, Tiruchirappalli, Tamil Nadu, India
sreelakshmikuttikod@gmail.com
V. Thota
Dept. of Mathematics, National Institute of Technology, Tiruchirappalli, Tamil Nadu, India
vamsinadh@nitt.edu
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-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.
Keywords: Equability, strong equability, quasi uniform rotundity, uniform rotundity, uniform strong proximinality, stability.
MSC: 41A65, 46B20; 46E40.
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