A Birkhoff-Orthogonal Dehghan--Rooin Type Constant in Banach Spaces

Abstract

For a real Banach space $X$. The Dehghan--Rooin magnitude compares the angular distance $\alpha [x,y]=\Vert x/\Vert x\Vert -y/\Vert y\Vert \Vert$ together with the skew angular distance $\beta [x,y]=\Vert x/\Vert y\Vert -y/\Vert x\Vert \Vert$. Motivated by the recent use of Birkhoff orthogonality in the Massera--Schaffer inequality, we introduce and study the Birkhoff-restricted Dehghan--Rooin constant \[ \operatorname{DR}_{B}(X)=\sup\left\{\frac{α[x,y]}{β[x,y]}:x,y\in X\setminus\{0\},\ x\perp_{B} y,\ β[x,y]\ne0\right\}. \] The restriction $x{\perp }_{B}y$ turns the global comparison of $\alpha$ and $\beta$ into a directional invariant which detects the interaction between supporting functionals and the radial normalization map. We obtain a scale-profile formula, sharp elementary bounds, stability under subspaces and ultrapowers, a variational characterization of the extremal case ${\mathrm{DR}}_B(X)=1$, estimates in terms of the classical Dehghan--Rooin constant, and consequences for uniformly convex and uniformly smooth spaces. In smooth two-dimensional spaces we derive an explicit formula through the normalized duality mapping, and in Radon planes we reduce the constant to a one-dimensional optimization on an arc of the unit circle. Several model computations are included, including Hilbert spaces, polyhedral planes, and two-dimensional $p$-norm spaces. The final section gives normal-structure type consequences through a companion Birkhoff angular modulus and lists several problems suggested by the new constant.