A study of local symmetry of Birkhoff-James orthogonality in Banach spaces

We present a complete characterization of the right-symmetric points in the one sum of two Banach spaces. We also obtain some basic properties of the left-symmetric (right-symmetric) points in the p sum, 1 <= p not equal 2 < infinity (1 < p not equal 2 < infinity), of two Banach spaces. Using these properties we (a) give examples of Banach spaces which do not have any non-zero left-symmetric points and (b) prove a complete characterization of those left-symmetric and right-symmetric points in the p sum, 1 < p not equal 2 < infinity, of two Banach spaces, whose components satisfy an additional norm assumption. We give examples of Banach spaces where all non-zero left-symmetric or right-symmetric points satisfy this additional norm assumption. We also present an alternative proof of the recently obtained characterization of the left-symmetric and the right-symmetric points in l(p)(n), n >= 3, and l(p), 1 <= p not equal 2 < infinity.