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James-type constants for Birkhoff--James orthogonality and approximate isosceles

  • Zhiyao Fang
  • Ying Xu
  • Qi Liu
  • Zhaohui Gu
  • Yongjin Li
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Abstract

We introduce two James-type constants associated with approximate isosceles and approximate Birkhoff--James orthogonality in real normed spaces. The isosceles family coincides with the classical James constant of Gao and Lau [J. Austral. Math. Soc. Ser. A 48 (1990), 101--112], while the Birkhoff family extends the orthogonal James constant of Baronti and Papini [Constr. Math. Anal. 5 (2022), no.~1, 37--45]. We show that approximate orthogonality constraints can, in certain settings, recover classical geometric constants exactly. In particular, the approximate isosceles family collapses to the classical James constant, while the approximate Birkhoff--James family forms a genuine parameter-dependent interpolation between the orthogonal and classical James constants.We also compare the Birkhoff profile with several classical geometric constants and study how it changes with the parameter.Several examples are also given to illustrate the relation between geometric constants and approximate orthogonality.

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Cite

BibTeXRIS
BibTeX
@misc{fang2026jamestype,
  title = {{James-type constants for Birkhoff--James orthogonality and approximate isosceles}},
  author = {Zhiyao Fang and Ying Xu and Qi Liu and Zhaohui Gu and Yongjin Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37502},
  eprint = {2609.37502},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37502},
}
RIS
TY  - UNPB
TI  - James-type constants for Birkhoff--James orthogonality and approximate isosceles
AU  - Zhiyao Fang
AU  - Ying Xu
AU  - Qi Liu
AU  - Zhaohui Gu
AU  - Yongjin Li
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.37502
UR  - https://arxiv.org/abs/2609.37502
AB  - We introduce two James-type constants associated with approximate isosceles and approximate Birkhoff--James orthogonality in real normed spaces. The isosceles family coincides with the classical James constant of Gao and Lau [J. Austral. Math. Soc. Ser. A 48 (1990), 101--112], while the Birkhoff family extends the orthogonal James constant of Baronti and Papini [Constr. Math. Anal. 5 (2022), no.~1, 37--45]. We show that approximate orthogonality constraints can, in certain settings, recover classical geometric constants exactly. In particular, the approximate isosceles family collapses to the classical James constant, while the approximate Birkhoff--James family forms a genuine parameter-dependent interpolation between the orthogonal and classical James constants.We also compare the Birkhoff profile with several classical geometric constants and study how it changes with the parameter.Several examples are also given to illustrate the relation between geometric constants and approximate orthogonality.
ER  -