Characterization of Approximate Birkhoff-James Orthogonality Sets in a Banach Space

There are two notions of approximate Birkhoff-James orthogonality sets in a Banach space. A geometric study of them has recently been done by D. Sain, K. Paul and A. Mal [On approximate Birkhoff-James orthogonality and normal cones in a normed space, J. Convex Analysis 26 (2019) 341--351]. We here characterize completely one of these sets in terms of normal cones. We obtain a uniqueness criterion between approximate orthogonality sets and normal cones in a Banach space. We also investigate the action of a bounded linear operator on approximate Birkhoff-James orthogonality sets.