Ball proximinal and strongly ball proximinal spaces
Let Y be an E-proximinal (respectively, a strongly proximinal) subspace of X. We prove that Y is (strongly) ball proximinal in X if and only if for any x € X with (x + Y) ∩ Bx ≈ θ, (x + Y) ∩ Bx is (strongly) proximinal in x + Y. Using this characterization and a smart construction, we obtain three Banach spaces Z ∩Y ∩ X such that Z is ball proximinal in X and Y/Z is ball proximinal in X/Z, but Y is not ball proximinal in X. This solves a problem raised by Bandyopadhyay, Lin, and Rao  in 2007.
Journal of Convex Analysis
Lin, P., Zhang, W., & Zheng, B. (2015). Ball proximinal and strongly ball proximinal spaces. Journal of Convex Analysis, 22 (3), 673-685. Retrieved from https://digitalcommons.memphis.edu/facpubs/4265