Orthogonality induced by norm derivatives: a new geometric constant and symmetry

In this article we study the difference between orthogonality induced by norm derivatives (known as \(\rho \)-orthogonality) and Birkhoff-James orthogonality in a normed linear space \(\mathbb {X}\) by introducing a new geometric constant, denoted by \(\Gamma (\mathbb {X}).\) We explore the relation between various geometric properties of the space and the constant \(\Gamma (\mathbb {X}).\) We also investigate the left symmetric and right symmetric elements of a normed linear space with respect to \(\rho \)-orthogonality and obtain a characterization of the same. We characterize inner product spaces among normed linear spaces using the symmetricity of \(\rho \)-orthogonality. Finally, we provide a complete description of both left symmetric and right symmetric elements with respect to \(\rho \)-orthogonality for some particular Banach spaces.