The Complex Banach Isometric Conjecture
Abstract
Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex (n+1)-space whose complex hyperplane sections are all complex-linearly equivalent. We show that such a body is a Hermitian ellipsoid. The proof adapts the recent bundle--degree method of Lu and Yang for the real problem, with two features specific to the complex setting. After covariance normalization, write G=Autℂ(S) for the complex-linear symmetry group of a model section S. The exact section maps form a principal G-bundle over S2n+1; reduction to G∘ places its obstruction in π2n(G∘), which is finite. Pulling back by a suitable positive-degree self-map trivializes this bundle and yields a global Lipschitz family of exact complex-linear section maps. Brouwer degree and a signed degree formula then imply that pS2n+2 and pS2n+4 are real homogeneous polynomials, where pS is the norm of a model section. Unique factorization forces pS2 to be quadratic, and phase invariance makes the resulting quadratic form Hermitian. The parallelogram identity then yields the general complex Banach-space statement.
Related
- A solution to Banach's isometric conjecture
- Rigidity for local holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains
- Non-maximal quasi-isometric PU2,n+1-representations via alternating surfaces in complex pseudo-hyperbolic spaces
- Embedding complexity into the universal Banach space and the strong Novikov conjecture
- Real alternative Daugavet property without Daugavet property in a separable complex Banach space
BibTeX
@misc{dai2026complex,
title = {{The Complex Banach Isometric Conjecture}},
author = {Xinan Dai and Wendao Deng and Yidong Shi and Tailin Wu and Yuchen Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26286},
eprint = {2609.26286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26286},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Complex Banach Isometric Conjecture
AU - Xinan Dai
AU - Wendao Deng
AU - Yidong Shi
AU - Tailin Wu
AU - Yuchen Yang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26286
UR - https://arxiv.org/abs/2609.26286
AB - Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex (n+1)-space whose complex hyperplane sections are all complex-linearly equivalent. We show that such a body is a Hermitian ellipsoid. The proof adapts the recent bundle--degree method of Lu and Yang for the real problem, with two features specific to the complex setting. After covariance normalization, write G=Autℂ(S) for the complex-linear symmetry group of a model section S. The exact section maps form a principal G-bundle over S^2n+1; reduction to G^∘ places its obstruction in π2n(G^∘), which is finite. Pulling back by a suitable positive-degree self-map trivializes this bundle and yields a global Lipschitz family of exact complex-linear section maps. Brouwer degree and a signed degree formula then imply that p_S^2n+2 and p_S^2n+4 are real homogeneous polynomials, where p_S is the norm of a model section. Unique factorization forces p_S^2 to be quadratic, and phase invariance makes the resulting quadratic form Hermitian. The parallelogram identity then yields the general complex Banach-space statement.
ER -