Isosceles-Orthogonal Takahashi–von Neumann–Jordan-Type Constants in Banach Spaces

We introduce an isosceles-orthogonal Takahashi–von Neumann–Jordan-type constant for real Banach spaces, defined through power means of chord lengths generated by isosceles-orthogonal unit vectors. We establish sharp estimates linking the associated profile and the new constant to the James constant, thereby clarifying their geometric relationship. In finite-dimensional spaces, we characterize the extremal case and prove a rigidity property that yields an exact criterion for failure of uniform non-squareness. We further derive an isosceles representation of Takahashi’s unrestricted profile and obtain exact values for classical ${\mathsf{\ell }}_p$ and $L_p[0,1]$ spaces for $-\infty \le t\le 2$. As an application to two-dimensional geometry, we prove a sharp universal estimate for Radon planes and characterize the equality case by affine regular hexagonal unit spheres. These results provide a unified framework connecting isosceles orthogonality, James-type geometry, and Takahashi-type constants.