  We study two notions of approximate Birkhoff-James orthogonality in a normed
space, from a geometric point of view, and characterize them in terms of normal
cones. We further explore the interconnection between normal cones and
approximate Birkhoff-James orthogonality to obtain a complete characterization
of normal cones in a two-dimensional smooth Banach space. We also obtain a
uniqueness theorem for approximate Birkhoff-James orthogonality set in a normed
space.
