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 α[x,y]=|x/|x|−y/|y|| together with the skew angular distance β[x,y]=|x/|y|−y/|x||. Motivated by the recent use of Birkhoff orthogonality in the Massera--Schaffer inequality, we introduce and study the Birkhoff-restricted Dehghan--Rooin constant DRB(X)=sup{α[x,y]/β[x,y]:x,y∈ X∖{0}, x⊥B y, β[x,y]≠0}. The restriction x⊥B y turns the global comparison of α and β 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 DRB(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.
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BibTeX
@misc{du2026birkhofforthogonal,
title = {{A Birkhoff-Orthogonal Dehghan--Rooin Type Constant in Banach Spaces}},
author = {Jia Du and Tingting Hu and Wenwen Zhang and Qi Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27754},
eprint = {2609.27754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27754},
}
RIS
TY - UNPB
TI - A Birkhoff-Orthogonal Dehghan--Rooin Type Constant in Banach Spaces
AU - Jia Du
AU - Tingting Hu
AU - Wenwen Zhang
AU - Qi Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27754
UR - https://arxiv.org/abs/2609.27754
AB - For a real Banach space X. The Dehghan--Rooin magnitude compares the angular distance α[x,y]=|x/|x|−y/|y|| together with the skew angular distance β[x,y]=|x/|y|−y/|x||. Motivated by the recent use of Birkhoff orthogonality in the Massera--Schaffer inequality, we introduce and study the Birkhoff-restricted Dehghan--Rooin constant DR_B(X)=sup{α[x,y]/β[x,y]:x,y∈ X∖{0}, x⊥_B y, β[x,y]≠0}. The restriction x⊥_B y turns the global comparison of α and β 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 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.
ER -