Some moduli related to \(\rho _{\pm }\)-orthogonalities and semi-orthogonality in Banach spaces

In this article, we introduce several new moduli of convexity connected with \(\rho _{\pm }\)-orthogonalities and semi-orthogonality, which are closely related to the modulus of convexity \(\delta _{X}(\varepsilon )\). In particular, these new parameters are computed for X being some specific spaces. Moreover, we give some applications of these two coefficients \(\delta _{\perp }(\rho _{\pm }, X)\) and investigate the relation between these two parameters and the approximate symmetry of \(\rho _{-}\)-orthogonality. In the meantime, we give characterization of the Radon plane with an affine-hexagonal unit sphere in terms of these new moduli of convexity. Moreover, we also consider the moduli of smoothness related to \(\rho _{\pm }\)-orthgonalities and semi-orthogonality. In the end, we discuss some applications of these new moduli of smoothness and study some relationships between these new parameters and uniform non-squareness, uniform convexity.