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

In this article we study the difference between orthogonality induced by the 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 $\mathrm{\Gamma }(\mathbb{X})\mathrm{.}$ We explore the relation between various geometric properties of the space and the constant $\mathrm{\Gamma }(\mathbb{X})\mathrm{.}$ 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.