Arc-length Orthogonality in Normed Linear Spaces

The existence and uniqueness properties of arc-length orthogonality and the existence and uniqueness of orthogonal diagonals in the sense of arc-length orthogonality are studied in normed linear spaces. By applying related results of homogeneous direction of isosceles orthogonality( Pythagorean orthogonality,resp.),the relation of arc-length orthogonality to isosceles orthogonality( Pythagorean orthogonality,resp) is studied. It is proved that if isosceles orthogonality( Pythagorean orthogonality,resp.) implies arc-length orthogonality or arc-length orthogonality implies isosceles orthogonality( Pythagorean orthogonality,resp.) in the intersection of a neighborhood of a fixed unit vector and the unit sphere in a real Banach space,then the underlying space is an inner product space.